Theorem isotr 2935

Description: Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33.

Assertion

Ref	Expression
isotr	⊢ ((H Isom R, S (A, B) ∧ G Isom S, T (B, C)) → (G ∘ H) Isom R, T (A, C))

Proof of Theorem isotr

Step	Hyp	Ref	Expression
1		pm3.26 256	. . . . . 6 ⊢ ((G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u))) → G:B–1-1-onto→C)
2		pm3.26 256	. . . . . 6 ⊢ ((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) → H:A–1-1-onto→B)
3	1, 2	anim12i 268	. . . . 5 ⊢ (((G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u))) ∧ (H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w)))) → (G:B–1-1-onto→C ∧ H:A–1-1-onto→B))
4	3	ancoms 334	. . . 4 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) → (G:B–1-1-onto→C ∧ H:A–1-1-onto→B))
5		f1oco 2816	. . . 4 ⊢ ((G:B–1-1-onto→C ∧ H:A–1-1-onto→B) → (G ∘ H):A–1-1-onto→C)
6	4, 5	syl 12	. . 3 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) → (G ∘ H):A–1-1-onto→C)
7		f1of 2800	. . . . . . . . . . 11 ⊢ (H:A–1-1-onto→B → H:A–→B)
8		ffvrn 2890	. . . . . . . . . . . . 13 ⊢ ((H:A–→B ∧ x ∈ A) → (H ‘x) ∈ B)
9	8	exp 291	. . . . . . . . . . . 12 ⊢ (H:A–→B → (x ∈ A → (H ‘x) ∈ B))
10		ffvrn 2890	. . . . . . . . . . . . 13 ⊢ ((H:A–→B ∧ y ∈ A) → (H ‘y) ∈ B)
11	10	exp 291	. . . . . . . . . . . 12 ⊢ (H:A–→B → (y ∈ A → (H ‘y) ∈ B))
12	9, 11	anim12d 431	. . . . . . . . . . 11 ⊢ (H:A–→B → ((x ∈ A ∧ y ∈ A) → ((H ‘x) ∈ B ∧ (H ‘y) ∈ B)))
13	7, 12	syl 12	. . . . . . . . . 10 ⊢ (H:A–1-1-onto→B → ((x ∈ A ∧ y ∈ A) → ((H ‘x) ∈ B ∧ (H ‘y) ∈ B)))
14	13	adantr 306	. . . . . . . . 9 ⊢ ((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) → ((x ∈ A ∧ y ∈ A) → ((H ‘x) ∈ B ∧ (H ‘y) ∈ B)))
15		breq1 2065	. . . . . . . . . . . 12 ⊢ (v = (H ‘x) → (vSu ↔ (H ‘x)Su))
16		fveq2 2832	. . . . . . . . . . . . 13 ⊢ (v = (H ‘x) → (G ‘v) = (G ‘(H ‘x)))
17	16	breq1d 2071	. . . . . . . . . . . 12 ⊢ (v = (H ‘x) → ((G ‘v)T(G ‘u) ↔ (G ‘(H ‘x))T(G ‘u)))
18	15, 17	bibi12d 477	. . . . . . . . . . 11 ⊢ (v = (H ‘x) → ((vSu ↔ (G ‘v)T(G ‘u)) ↔ ((H ‘x)Su ↔ (G ‘(H ‘x))T(G ‘u))))
19		breq2 2066	. . . . . . . . . . . 12 ⊢ (u = (H ‘y) → ((H ‘x)Su ↔ (H ‘x)S(H ‘y)))
20		fveq2 2832	. . . . . . . . . . . . 13 ⊢ (u = (H ‘y) → (G ‘u) = (G ‘(H ‘y)))
21	20	breq2d 2072	. . . . . . . . . . . 12 ⊢ (u = (H ‘y) → ((G ‘(H ‘x))T(G ‘u) ↔ (G ‘(H ‘x))T(G ‘(H ‘y))))
22	19, 21	bibi12d 477	. . . . . . . . . . 11 ⊢ (u = (H ‘y) → (((H ‘x)Su ↔ (G ‘(H ‘x))T(G ‘u)) ↔ ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y)))))
23	18, 22	rcla42v 1404	. . . . . . . . . 10 ⊢ (∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)) → (((H ‘x) ∈ B ∧ (H ‘y) ∈ B) → ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y)))))
24	23	adantl 305	. . . . . . . . 9 ⊢ ((G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u))) → (((H ‘x) ∈ B ∧ (H ‘y) ∈ B) → ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y)))))
25	14, 24	sylan9 359	. . . . . . . 8 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) → ((x ∈ A ∧ y ∈ A) → ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y)))))
26	25	imp 277	. . . . . . 7 ⊢ ((((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) ∧ (x ∈ A ∧ y ∈ A)) → ((H ‘x)S(H ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y))))
27		breq1 2065	. . . . . . . . . . . 12 ⊢ (z = x → (zRw ↔ xRw))
28		fveq2 2832	. . . . . . . . . . . . 13 ⊢ (z = x → (H ‘z) = (H ‘x))
29	28	breq1d 2071	. . . . . . . . . . . 12 ⊢ (z = x → ((H ‘z)S(H ‘w) ↔ (H ‘x)S(H ‘w)))
30	27, 29	bibi12d 477	. . . . . . . . . . 11 ⊢ (z = x → ((zRw ↔ (H ‘z)S(H ‘w)) ↔ (xRw ↔ (H ‘x)S(H ‘w))))
31		breq2 2066	. . . . . . . . . . . 12 ⊢ (w = y → (xRw ↔ xRy))
32		fveq2 2832	. . . . . . . . . . . . 13 ⊢ (w = y → (H ‘w) = (H ‘y))
33	32	breq2d 2072	. . . . . . . . . . . 12 ⊢ (w = y → ((H ‘x)S(H ‘w) ↔ (H ‘x)S(H ‘y)))
34	31, 33	bibi12d 477	. . . . . . . . . . 11 ⊢ (w = y → ((xRw ↔ (H ‘x)S(H ‘w)) ↔ (xRy ↔ (H ‘x)S(H ‘y))))
35	30, 34	rcla42v 1404	. . . . . . . . . 10 ⊢ (∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w)) → ((x ∈ A ∧ y ∈ A) → (xRy ↔ (H ‘x)S(H ‘y))))
36	35	imp 277	. . . . . . . . 9 ⊢ ((∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w)) ∧ (x ∈ A ∧ y ∈ A)) → (xRy ↔ (H ‘x)S(H ‘y)))
37	36	adantll 309	. . . . . . . 8 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (x ∈ A ∧ y ∈ A)) → (xRy ↔ (H ‘x)S(H ‘y)))
38	37	adantlr 310	. . . . . . 7 ⊢ ((((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) ∧ (x ∈ A ∧ y ∈ A)) → (xRy ↔ (H ‘x)S(H ‘y)))
39		fvco3 2867	. . . . . . . . . . 11 ⊢ (((Fun G ∧ H:A–→B) ∧ x ∈ A) → ((G ∘ H) ‘x) = (G ‘(H ‘x)))
40	39	adantrr 312	. . . . . . . . . 10 ⊢ (((Fun G ∧ H:A–→B) ∧ (x ∈ A ∧ y ∈ A)) → ((G ∘ H) ‘x) = (G ‘(H ‘x)))
41		fvco3 2867	. . . . . . . . . . 11 ⊢ (((Fun G ∧ H:A–→B) ∧ y ∈ A) → ((G ∘ H) ‘y) = (G ‘(H ‘y)))
42	41	adantrl 311	. . . . . . . . . 10 ⊢ (((Fun G ∧ H:A–→B) ∧ (x ∈ A ∧ y ∈ A)) → ((G ∘ H) ‘y) = (G ‘(H ‘y)))
43	40, 42	breq12d 2073	. . . . . . . . 9 ⊢ (((Fun G ∧ H:A–→B) ∧ (x ∈ A ∧ y ∈ A)) → (((G ∘ H) ‘x)T((G ∘ H) ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y))))
44		f1ofun 2802	. . . . . . . . . 10 ⊢ (G:B–1-1-onto→C → Fun G)
45	44, 7	anim12i 268	. . . . . . . . 9 ⊢ ((G:B–1-1-onto→C ∧ H:A–1-1-onto→B) → (Fun G ∧ H:A–→B))
46	43, 45	sylan 343	. . . . . . . 8 ⊢ (((G:B–1-1-onto→C ∧ H:A–1-1-onto→B) ∧ (x ∈ A ∧ y ∈ A)) → (((G ∘ H) ‘x)T((G ∘ H) ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y))))
47	46, 4	sylan 343	. . . . . . 7 ⊢ ((((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) ∧ (x ∈ A ∧ y ∈ A)) → (((G ∘ H) ‘x)T((G ∘ H) ‘y) ↔ (G ‘(H ‘x))T(G ‘(H ‘y))))
48	26, 38, 47	3bitr4d 424	. . . . . 6 ⊢ ((((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) ∧ (x ∈ A ∧ y ∈ A)) → (xRy ↔ ((G ∘ H) ‘x)T((G ∘ H) ‘y)))
49	48	exp32 294	. . . . 5 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) → (x ∈ A → (y ∈ A → (xRy ↔ ((G ∘ H) ‘x)T((G ∘ H) ‘y)))))
50	49	r19.21adv 1262	. . . 4 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) → (x ∈ A → ∀y ∈ A (xRy ↔ ((G ∘ H) ‘x)T((G ∘ H) ‘y))))
51	50	r19.21aiv 1259	. . 3 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) → ∀x ∈ A ∀y ∈ A (xRy ↔ ((G ∘ H) ‘x)T((G ∘ H) ‘y)))
52	6, 51	jca 236	. 2 ⊢ (((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))) → ((G ∘ H):A–1-1-onto→C ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ ((G ∘ H) ‘x)T((G ∘ H) ‘y))))
53		df-iso 2439	. . 3 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))))
54		df-iso 2439	. . 3 ⊢ (G Isom S, T (B, C) ↔ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u))))
55	53, 54	anbi12i 369	. 2 ⊢ ((H Isom R, S (A, B) ∧ G Isom S, T (B, C)) ↔ ((H:A–1-1-onto→B ∧ ∀z ∈ A ∀w ∈ A (zRw ↔ (H ‘z)S(H ‘w))) ∧ (G:B–1-1-onto→C ∧ ∀v ∈ B ∀u ∈ B (vSu ↔ (G ‘v)T(G ‘u)))))
56		df-iso 2439	. 2 ⊢ ((G ∘ H) Isom R, T (A, C) ↔ ((G ∘ H):A–1-1-onto→C ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ ((G ∘ H) ‘x)T((G ∘ H) ‘y))))
57	52, 55, 56	3imtr4 192	1 ⊢ ((H Isom R, S (A, B) ∧ G Isom S, T (B, C)) → (G ∘ H) Isom R, T (A, C))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054   ∘ ccom 2414  Fun wfun 2416  –→wf 2418  –1-1-onto→wf1o 2421   ‘cfv 2422   Isom wiso 2423

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439

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