Theorem addcmpblnr 3975

Description: Lemma showing compatibility of addition.

Hypotheses

Ref	Expression
cmpblnr.1	⊢ A ∈ V
cmpblnr.2	⊢ B ∈ V
cmpblnr.3	⊢ C ∈ V
cmpblnr.4	⊢ D ∈ V
cmpblnr.5	⊢ F ∈ V
cmpblnr.6	⊢ G ∈ V
cmpblnr.7	⊢ R ∈ V
cmpblnr.8	⊢ S ∈ V

Assertion

Ref	Expression
addcmpblnr	⊢ ((((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ P)) ∧ ((F ∈ P ∧ G ∈ P) ∧ (R ∈ P ∧ S ∈ P))) → (((A +P D) = (B +P C) ∧ (F +P S) = (G +P R)) → ⟨(A +P F), (B +P G)⟩ ~R ⟨(C +P R), (D +P S)⟩))

Proof of Theorem addcmpblnr

Step	Hyp	Ref	Expression
1		addclpr 3914	. . . . . . . 8 ⊢ ((A ∈ P ∧ F ∈ P) → (A +P F) ∈ P)
2		addclpr 3914	. . . . . . . 8 ⊢ ((B ∈ P ∧ G ∈ P) → (B +P G) ∈ P)
3	1, 2	anim12i 268	. . . . . . 7 ⊢ (((A ∈ P ∧ F ∈ P) ∧ (B ∈ P ∧ G ∈ P)) → ((A +P F) ∈ P ∧ (B +P G) ∈ P))
4	3	an4s 390	. . . . . 6 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (F ∈ P ∧ G ∈ P)) → ((A +P F) ∈ P ∧ (B +P G) ∈ P))
5		addclpr 3914	. . . . . . . 8 ⊢ ((C ∈ P ∧ R ∈ P) → (C +P R) ∈ P)
6		addclpr 3914	. . . . . . . 8 ⊢ ((D ∈ P ∧ S ∈ P) → (D +P S) ∈ P)
7	5, 6	anim12i 268	. . . . . . 7 ⊢ (((C ∈ P ∧ R ∈ P) ∧ (D ∈ P ∧ S ∈ P)) → ((C +P R) ∈ P ∧ (D +P S) ∈ P))
8	7	an4s 390	. . . . . 6 ⊢ (((C ∈ P ∧ D ∈ P) ∧ (R ∈ P ∧ S ∈ P)) → ((C +P R) ∈ P ∧ (D +P S) ∈ P))
9	4, 8	anim12i 268	. . . . 5 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (F ∈ P ∧ G ∈ P)) ∧ ((C ∈ P ∧ D ∈ P) ∧ (R ∈ P ∧ S ∈ P))) → (((A +P F) ∈ P ∧ (B +P G) ∈ P) ∧ ((C +P R) ∈ P ∧ (D +P S) ∈ P)))
10	9	an4s 390	. . . 4 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ P)) ∧ ((F ∈ P ∧ G ∈ P) ∧ (R ∈ P ∧ S ∈ P))) → (((A +P F) ∈ P ∧ (B +P G) ∈ P) ∧ ((C +P R) ∈ P ∧ (D +P S) ∈ P)))
11		enrbreq 3968	. . . 4 ⊢ ((((A +P F) ∈ P ∧ (B +P G) ∈ P) ∧ ((C +P R) ∈ P ∧ (D +P S) ∈ P)) → (⟨(A +P F), (B +P G)⟩ ~R ⟨(C +P R), (D +P S)⟩ ↔ ((A +P F) +P (D +P S)) = ((B +P G) +P (C +P R))))
12	10, 11	syl 12	. . 3 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ P)) ∧ ((F ∈ P ∧ G ∈ P) ∧ (R ∈ P ∧ S ∈ P))) → (⟨(A +P F), (B +P G)⟩ ~R ⟨(C +P R), (D +P S)⟩ ↔ ((A +P F) +P (D +P S)) = ((B +P G) +P (C +P R))))
13		cmpblnr.1	. . . . 5 ⊢ A ∈ V
14		cmpblnr.5	. . . . 5 ⊢ F ∈ V
15		cmpblnr.4	. . . . 5 ⊢ D ∈ V
16		visset 1350	. . . . . 6 ⊢ x ∈ V
17		visset 1350	. . . . . 6 ⊢ y ∈ V
18	16, 17	addcompr 3917	. . . . 5 ⊢ (x +P y) = (y +P x)
19		visset 1350	. . . . . 6 ⊢ z ∈ V
20	17, 19	addasspr 3918	. . . . 5 ⊢ ((x +P y) +P z) = (x +P (y +P z))
21		cmpblnr.8	. . . . 5 ⊢ S ∈ V
22	13, 14, 15, 18, 20, 21	caopr4 3078	. . . 4 ⊢ ((A +P F) +P (D +P S)) = ((A +P D) +P (F +P S))
23		cmpblnr.2	. . . . 5 ⊢ B ∈ V
24		cmpblnr.6	. . . . 5 ⊢ G ∈ V
25		cmpblnr.3	. . . . 5 ⊢ C ∈ V
26		cmpblnr.7	. . . . 5 ⊢ R ∈ V
27	23, 24, 25, 18, 20, 26	caopr4 3078	. . . 4 ⊢ ((B +P G) +P (C +P R)) = ((B +P C) +P (G +P R))
28	22, 27	cleq12i 1114	. . 3 ⊢ (((A +P F) +P (D +P S)) = ((B +P G) +P (C +P R)) ↔ ((A +P D) +P (F +P S)) = ((B +P C) +P (G +P R)))
29	12, 28	syl6bb 414	. 2 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ P)) ∧ ((F ∈ P ∧ G ∈ P) ∧ (R ∈ P ∧ S ∈ P))) → (⟨(A +P F), (B +P G)⟩ ~R ⟨(C +P R), (D +P S)⟩ ↔ ((A +P D) +P (F +P S)) = ((B +P C) +P (G +P R))))
30		opreq12 3008	. 2 ⊢ (((A +P D) = (B +P C) ∧ (F +P S) = (G +P R)) → ((A +P D) +P (F +P S)) = ((B +P C) +P (G +P R)))
31	29, 30	syl5bir 184	1 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (C ∈ P ∧ D ∈ P)) ∧ ((F ∈ P ∧ G ∈ P) ∧ (R ∈ P ∧ S ∈ P))) → (((A +P D) = (B +P C) ∧ (F +P S) = (G +P R)) → ⟨(A +P F), (B +P G)⟩ ~R ⟨(C +P R), (D +P S)⟩))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  (class class class)co 3001  Pcnp 3779   +_P cpp 3781   ~_R cer 3786

This theorem is referenced by:  addsrpr 3978

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  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  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-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  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-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-plp 3882  df-enr 3960

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