Theorem oawordri 3152

Description: Weak ordering property of ordinal addition. Proposition 8.7 of [TakeutiZaring] p. 59.

Assertion

Ref	Expression
oawordri	|- ((A e. On /\ B e. On /\ C e. On) -> (A (_ B -> (A +o C) (_ (B +o C)))

Proof of Theorem oawordri

Step	Hyp	Ref	Expression
1		opreq2 3007	. . . . . 6 |- (x = (/) -> (A +o x) = (A +o (/)))
2		opreq2 3007	. . . . . 6 |- (x = (/) -> (B +o x) = (B +o (/)))
3	1, 2	sseq12d 1529	. . . . 5 |- (x = (/) -> ((A +o x) (_ (B +o x) <-> (A +o (/)) (_ (B +o (/))))
4		opreq2 3007	. . . . . 6 |- (x = y -> (A +o x) = (A +o y))
5		opreq2 3007	. . . . . 6 |- (x = y -> (B +o x) = (B +o y))
6	4, 5	sseq12d 1529	. . . . 5 |- (x = y -> ((A +o x) (_ (B +o x) <-> (A +o y) (_ (B +o y)))
7		opreq2 3007	. . . . . 6 |- (x = suc y -> (A +o x) = (A +o suc y))
8		opreq2 3007	. . . . . 6 |- (x = suc y -> (B +o x) = (B +o suc y))
9	7, 8	sseq12d 1529	. . . . 5 |- (x = suc y -> ((A +o x) (_ (B +o x) <-> (A +o suc y) (_ (B +o suc y)))
10		opreq2 3007	. . . . . 6 |- (x = C -> (A +o x) = (A +o C))
11		opreq2 3007	. . . . . 6 |- (x = C -> (B +o x) = (B +o C))
12	10, 11	sseq12d 1529	. . . . 5 |- (x = C -> ((A +o x) (_ (B +o x) <-> (A +o C) (_ (B +o C)))
13		oa0 3124	. . . . . . . 8 |- (A e. On -> (A +o (/)) = A)
14	13	adantr 306	. . . . . . 7 |- ((A e. On /\ B e. On) -> (A +o (/)) = A)
15		oa0 3124	. . . . . . . 8 |- (B e. On -> (B +o (/)) = B)
16	15	adantl 305	. . . . . . 7 |- ((A e. On /\ B e. On) -> (B +o (/)) = B)
17	14, 16	sseq12d 1529	. . . . . 6 |- ((A e. On /\ B e. On) -> ((A +o (/)) (_ (B +o (/)) <-> A (_ B))
18	17	biimpar 325	. . . . 5 |- (((A e. On /\ B e. On) /\ A (_ B) -> (A +o (/)) (_ (B +o (/)))
19		ordsucsssuc 2325	. . . . . . . . . . . 12 |- ((Ord (A +o y) /\ Ord (B +o y)) -> ((A +o y) (_ (B +o y) <-> suc (A +o y) (_ suc (B +o y)))
20		oacl 3138	. . . . . . . . . . . . 13 |- ((A e. On /\ y e. On) -> (A +o y) e. On)
21		eloni 2209	. . . . . . . . . . . . 13 |- ((A +o y) e. On -> Ord (A +o y))
22	20, 21	syl 12	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> Ord (A +o y))
23		oacl 3138	. . . . . . . . . . . . 13 |- ((B e. On /\ y e. On) -> (B +o y) e. On)
24		eloni 2209	. . . . . . . . . . . . 13 |- ((B +o y) e. On -> Ord (B +o y))
25	23, 24	syl 12	. . . . . . . . . . . 12 |- ((B e. On /\ y e. On) -> Ord (B +o y))
26	19, 22, 25	syl2an 349	. . . . . . . . . . 11 |- (((A e. On /\ y e. On) /\ (B e. On /\ y e. On)) -> ((A +o y) (_ (B +o y) <-> suc (A +o y) (_ suc (B +o y)))
27	26	anandirs 395	. . . . . . . . . 10 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o y) (_ (B +o y) <-> suc (A +o y) (_ suc (B +o y)))
28		oasuc 3131	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> (A +o suc y) = suc (A +o y))
29	28	adantlr 310	. . . . . . . . . . 11 |- (((A e. On /\ B e. On) /\ y e. On) -> (A +o suc y) = suc (A +o y))
30		oasuc 3131	. . . . . . . . . . . 12 |- ((B e. On /\ y e. On) -> (B +o suc y) = suc (B +o y))
31	30	adantll 309	. . . . . . . . . . 11 |- (((A e. On /\ B e. On) /\ y e. On) -> (B +o suc y) = suc (B +o y))
32	29, 31	sseq12d 1529	. . . . . . . . . 10 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o suc y) (_ (B +o suc y) <-> suc (A +o y) (_ suc (B +o y)))
33	27, 32	bitr4d 409	. . . . . . . . 9 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o y) (_ (B +o y) <-> (A +o suc y) (_ (B +o suc y)))
34	33	biimpd 135	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ y e. On) -> ((A +o y) (_ (B +o y) -> (A +o suc y) (_ (B +o suc y)))
35	34	exp 291	. . . . . . 7 |- ((A e. On /\ B e. On) -> (y e. On -> ((A +o y) (_ (B +o y) -> (A +o suc y) (_ (B +o suc y))))
36	35	com12 13	. . . . . 6 |- (y e. On -> ((A e. On /\ B e. On) -> ((A +o y) (_ (B +o y) -> (A +o suc y) (_ (B +o suc y))))
37	36	adantrd 308	. . . . 5 |- (y e. On -> (((A e. On /\ B e. On) /\ A (_ B) -> ((A +o y) (_ (B +o y) -> (A +o suc y) (_ (B +o suc y))))
38		visset 1350	. . . . . . . . 9 |- x e. V
39		oalim 3135	. . . . . . . . . . . 12 |- ((A e. On /\ (x e. V /\ Lim x)) -> (A +o x) = U.y e. x (A +o y))
40	39	adantlr 310	. . . . . . . . . . 11 |- (((A e. On /\ B e. On) /\ (x e. V /\ Lim x)) -> (A +o x) = U.y e. x (A +o y))
41		oalim 3135	. . . . . . . . . . . 12 |- ((B e. On /\ (x e. V /\ Lim x)) -> (B +o x) = U.y e. x (B +o y))
42	41	adantll 309	. . . . . . . . . . 11 |- (((A e. On /\ B e. On) /\ (x e. V /\ Lim x)) -> (B +o x) = U.y e. x (B +o y))
43	40, 42	sseq12d 1529	. . . . . . . . . 10 |- (((A e. On /\ B e. On) /\ (x e. V /\ Lim x)) -> ((A +o x) (_ (B +o x) <-> U.y e. x (A +o y) (_ U.y e. x (B +o y)))
44		ss2iun 2005	. . . . . . . . . 10 |- (A.y e. x (A +o y) (_ (B +o y) -> U.y e. x (A +o y) (_ U.y e. x (B +o y))
45	43, 44	syl5bir 184	. . . . . . . . 9 |- (((A e. On /\ B e. On) /\ (x e. V /\ Lim x)) -> (A.y e. x (A +o y) (_ (B +o y) -> (A +o x) (_ (B +o x)))
46	38, 45	mpan21 531	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ Lim x) -> (A.y e. x (A +o y) (_ (B +o y) -> (A +o x) (_ (B +o x)))
47	46	ancoms 334	. . . . . . 7 |- ((Lim x /\ (A e. On /\ B e. On)) -> (A.y e. x (A +o y) (_ (B +o y) -> (A +o x) (_ (B +o x)))
48	47	exp 291	. . . . . 6 |- (Lim x -> ((A e. On /\ B e. On) -> (A.y e. x (A +o y) (_ (B +o y) -> (A +o x) (_ (B +o x))))
49	48	adantrd 308	. . . . 5 |- (Lim x -> (((A e. On /\ B e. On) /\ A (_ B) -> (A.y e. x (A +o y) (_ (B +o y) -> (A +o x) (_ (B +o x))))
50	3, 6, 9, 12, 18, 37, 49	tfinds3 2406	. . . 4 |- (C e. On -> (((A e. On /\ B e. On) /\ A (_ B) -> (A +o C) (_ (B +o C)))
51	50	exp4c 297	. . 3 |- (C e. On -> (A e. On -> (B e. On -> (A (_ B -> (A +o C) (_ (B +o C)))))
52	51	com3l 34	. 2 |- (A e. On -> (B e. On -> (C e. On -> (A (_ B -> (A +o C) (_ (B +o C)))))
53	52	3imp 608	1 |- ((A e. On /\ B e. On /\ C e. On) -> (A (_ B -> (A +o C) (_ (B +o C)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581   = weq 797   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348   (_ wss 1487  (/)c0 1707  U.ciun 1994  Ord word 2198  Oncon0 2199  Lim wlim 2200  suc csuc 2201  (class class class)co 3001   +o coa 3101

This theorem is referenced by:  oaword2 3155

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-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-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  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-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-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-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106

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