Theorem oeoelem 5273

Description: Lemma for oeoe 5274.

Hypotheses

Ref	Expression
oeoelem.1	|- A e. On
oeoelem.2	|- (/) e. A

Assertion

Ref	Expression
oeoelem	|- ((B e. On /\ C e. On) -> ((A ^o B) ^o C) = (A ^o (B .o C)))

Proof of Theorem oeoelem

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . 4 |- (x = (/) -> ((A ^o B) ^o x) = ((A ^o B) ^o (/)))
2		opreq2 4890	. . . . 5 |- (x = (/) -> (B .o x) = (B .o (/)))
3	2	opreq2d 4898	. . . 4 |- (x = (/) -> (A ^o (B .o x)) = (A ^o (B .o (/))))
4	1, 3	eqeq12d 1899	. . 3 |- (x = (/) -> (((A ^o B) ^o x) = (A ^o (B .o x)) <-> ((A ^o B) ^o (/)) = (A ^o (B .o (/)))))
5		opreq2 4890	. . . 4 |- (x = y -> ((A ^o B) ^o x) = ((A ^o B) ^o y))
6		opreq2 4890	. . . . 5 |- (x = y -> (B .o x) = (B .o y))
7	6	opreq2d 4898	. . . 4 |- (x = y -> (A ^o (B .o x)) = (A ^o (B .o y)))
8	5, 7	eqeq12d 1899	. . 3 |- (x = y -> (((A ^o B) ^o x) = (A ^o (B .o x)) <-> ((A ^o B) ^o y) = (A ^o (B .o y))))
9		opreq2 4890	. . . 4 |- (x = suc y -> ((A ^o B) ^o x) = ((A ^o B) ^o suc y))
10		opreq2 4890	. . . . 5 |- (x = suc y -> (B .o x) = (B .o suc y))
11	10	opreq2d 4898	. . . 4 |- (x = suc y -> (A ^o (B .o x)) = (A ^o (B .o suc y)))
12	9, 11	eqeq12d 1899	. . 3 |- (x = suc y -> (((A ^o B) ^o x) = (A ^o (B .o x)) <-> ((A ^o B) ^o suc y) = (A ^o (B .o suc y))))
13		opreq2 4890	. . . 4 |- (x = C -> ((A ^o B) ^o x) = ((A ^o B) ^o C))
14		opreq2 4890	. . . . 5 |- (x = C -> (B .o x) = (B .o C))
15	14	opreq2d 4898	. . . 4 |- (x = C -> (A ^o (B .o x)) = (A ^o (B .o C)))
16	13, 15	eqeq12d 1899	. . 3 |- (x = C -> (((A ^o B) ^o x) = (A ^o (B .o x)) <-> ((A ^o B) ^o C) = (A ^o (B .o C))))
17		oeoelem.1	. . . . . 6 |- A e. On
18		oecl 5218	. . . . . 6 |- ((A e. On /\ B e. On) -> (A ^o B) e. On)
19	17, 18	mpan 759	. . . . 5 |- (B e. On -> (A ^o B) e. On)
20		oe0 5206	. . . . 5 |- ((A ^o B) e. On -> ((A ^o B) ^o (/)) = 1o)
21	19, 20	syl 12	. . . 4 |- (B e. On -> ((A ^o B) ^o (/)) = 1o)
22		om0 5201	. . . . . 6 |- (B e. On -> (B .o (/)) = (/))
23	22	opreq2d 4898	. . . . 5 |- (B e. On -> (A ^o (B .o (/))) = (A ^o (/)))
24		oe0 5206	. . . . . 6 |- (A e. On -> (A ^o (/)) = 1o)
25	17, 24	ax-mp 7	. . . . 5 |- (A ^o (/)) = 1o
26	23, 25	syl6eq 1944	. . . 4 |- (B e. On -> (A ^o (B .o (/))) = 1o)
27	21, 26	eqtr4d 1928	. . 3 |- (B e. On -> ((A ^o B) ^o (/)) = (A ^o (B .o (/))))
28		oesuc 5211	. . . . . . 7 |- (((A ^o B) e. On /\ y e. On) -> ((A ^o B) ^o suc y) = (((A ^o B) ^o y) .o (A ^o B)))
29	28, 19	sylan 497	. . . . . 6 |- ((B e. On /\ y e. On) -> ((A ^o B) ^o suc y) = (((A ^o B) ^o y) .o (A ^o B)))
30		omsuc 5210	. . . . . . . 8 |- ((B e. On /\ y e. On) -> (B .o suc y) = ((B .o y) +o B))
31	30	opreq2d 4898	. . . . . . 7 |- ((B e. On /\ y e. On) -> (A ^o (B .o suc y)) = (A ^o ((B .o y) +o B)))
32		oeoa 5272	. . . . . . . . . 10 |- ((A e. On /\ (B .o y) e. On /\ B e. On) -> (A ^o ((B .o y) +o B)) = ((A ^o (B .o y)) .o (A ^o B)))
33	17, 32	mp3an1 1178	. . . . . . . . 9 |- (((B .o y) e. On /\ B e. On) -> (A ^o ((B .o y) +o B)) = ((A ^o (B .o y)) .o (A ^o B)))
34		omcl 5216	. . . . . . . . 9 |- ((B e. On /\ y e. On) -> (B .o y) e. On)
35	33, 34	sylan 497	. . . . . . . 8 |- (((B e. On /\ y e. On) /\ B e. On) -> (A ^o ((B .o y) +o B)) = ((A ^o (B .o y)) .o (A ^o B)))
36	35	anabss1 557	. . . . . . 7 |- ((B e. On /\ y e. On) -> (A ^o ((B .o y) +o B)) = ((A ^o (B .o y)) .o (A ^o B)))
37	31, 36	eqtrd 1925	. . . . . 6 |- ((B e. On /\ y e. On) -> (A ^o (B .o suc y)) = ((A ^o (B .o y)) .o (A ^o B)))
38	29, 37	eqeq12d 1899	. . . . 5 |- ((B e. On /\ y e. On) -> (((A ^o B) ^o suc y) = (A ^o (B .o suc y)) <-> (((A ^o B) ^o y) .o (A ^o B)) = ((A ^o (B .o y)) .o (A ^o B))))
39		opreq1 4889	. . . . 5 |- (((A ^o B) ^o y) = (A ^o (B .o y)) -> (((A ^o B) ^o y) .o (A ^o B)) = ((A ^o (B .o y)) .o (A ^o B)))
40	38, 39	syl5bir 227	. . . 4 |- ((B e. On /\ y e. On) -> (((A ^o B) ^o y) = (A ^o (B .o y)) -> ((A ^o B) ^o suc y) = (A ^o (B .o suc y))))
41	40	expcom 403	. . 3 |- (y e. On -> (B e. On -> (((A ^o B) ^o y) = (A ^o (B .o y)) -> ((A ^o B) ^o suc y) = (A ^o (B .o suc y)))))
42		visset 2295	. . . . . . 7 |- x e. _V
43		oelim 5214	. . . . . . . . . . 11 |- ((((A ^o B) e. On /\ (x e. _V /\ Lim x)) /\ (/) e. (A ^o B)) -> ((A ^o B) ^o x) = U_y e. x ((A ^o B) ^o y))
44	43, 18	sylanl1 509	. . . . . . . . . 10 |- ((((A e. On /\ B e. On) /\ (x e. _V /\ Lim x)) /\ (/) e. (A ^o B)) -> ((A ^o B) ^o x) = U_y e. x ((A ^o B) ^o y))
45		oeoelem.2	. . . . . . . . . . 11 |- (/) e. A
46		oen0 5261	. . . . . . . . . . 11 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (/) e. (A ^o B))
47	45, 46	mpan2 760	. . . . . . . . . 10 |- ((A e. On /\ B e. On) -> (/) e. (A ^o B))
48	44, 47	sylan2 500	. . . . . . . . 9 |- ((((A e. On /\ B e. On) /\ (x e. _V /\ Lim x)) /\ (A e. On /\ B e. On)) -> ((A ^o B) ^o x) = U_y e. x ((A ^o B) ^o y))
49	48	anabss1 557	. . . . . . . 8 |- (((A e. On /\ B e. On) /\ (x e. _V /\ Lim x)) -> ((A ^o B) ^o x) = U_y e. x ((A ^o B) ^o y))
50	17, 49	mpanl1 770	. . . . . . 7 |- ((B e. On /\ (x e. _V /\ Lim x)) -> ((A ^o B) ^o x) = U_y e. x ((A ^o B) ^o y))
51	42, 50	mpanr1 774	. . . . . 6 |- ((B e. On /\ Lim x) -> ((A ^o B) ^o x) = U_y e. x ((A ^o B) ^o y))
52		omlim 5213	. . . . . . . . 9 |- ((B e. On /\ (x e. _V /\ Lim x)) -> (B .o x) = U_y e. x (B .o y))
53	42, 52	mpanr1 774	. . . . . . . 8 |- ((B e. On /\ Lim x) -> (B .o x) = U_y e. x (B .o y))
54	53	opreq2d 4898	. . . . . . 7 |- ((B e. On /\ Lim x) -> (A ^o (B .o x)) = (A ^o U_y e. x (B .o y)))
55	42	a1i 8	. . . . . . . 8 |- ((B e. On /\ Lim x) -> x e. _V)
56		ordelon 3682	. . . . . . . . . . . 12 |- ((Ord x /\ y e. x) -> y e. On)
57		limord 3723	. . . . . . . . . . . 12 |- (Lim x -> Ord x)
58	56, 57	sylan 497	. . . . . . . . . . 11 |- ((Lim x /\ y e. x) -> y e. On)
59	34, 58	sylan2 500	. . . . . . . . . 10 |- ((B e. On /\ (Lim x /\ y e. x)) -> (B .o y) e. On)
60	59	anassrs 489	. . . . . . . . 9 |- (((B e. On /\ Lim x) /\ y e. x) -> (B .o y) e. On)
61	60	r19.21aiva 2176	. . . . . . . 8 |- ((B e. On /\ Lim x) -> A.y e. x (B .o y) e. On)
62		0ellim 3726	. . . . . . . . . 10 |- (Lim x -> (/) e. x)
63		ne0i 2881	. . . . . . . . . 10 |- ((/) e. x -> x =/= (/))
64	62, 63	syl 12	. . . . . . . . 9 |- (Lim x -> x =/= (/))
65	64	adantl 424	. . . . . . . 8 |- ((B e. On /\ Lim x) -> x =/= (/))
66		visset 2295	. . . . . . . . . 10 |- w e. _V
67		oelim 5214	. . . . . . . . . . . 12 |- (((A e. On /\ (w e. _V /\ Lim w)) /\ (/) e. A) -> (A ^o w) = U_z e. w (A ^o z))
68	45, 67	mpan2 760	. . . . . . . . . . 11 |- ((A e. On /\ (w e. _V /\ Lim w)) -> (A ^o w) = U_z e. w (A ^o z))
69	17, 68	mpan 759	. . . . . . . . . 10 |- ((w e. _V /\ Lim w) -> (A ^o w) = U_z e. w (A ^o z))
70	66, 69	mpan 759	. . . . . . . . 9 |- (Lim w -> (A ^o w) = U_z e. w (A ^o z))
71		oewordi 5266	. . . . . . . . . . . 12 |- (((z e. On /\ w e. On /\ A e. On) /\ (/) e. A) -> (z C_ w -> (A ^o z) C_ (A ^o w)))
72	45, 71	mpan2 760	. . . . . . . . . . 11 |- ((z e. On /\ w e. On /\ A e. On) -> (z C_ w -> (A ^o z) C_ (A ^o w)))
73	17, 72	mp3an3 1180	. . . . . . . . . 10 |- ((z e. On /\ w e. On) -> (z C_ w -> (A ^o z) C_ (A ^o w)))
74	73	3impia 1064	. . . . . . . . 9 |- ((z e. On /\ w e. On /\ z C_ w) -> (A ^o z) C_ (A ^o w))
75	70, 74	onopriun 5118	. . . . . . . 8 |- ((x e. _V /\ A.y e. x (B .o y) e. On /\ x =/= (/)) -> (A ^o U_y e. x (B .o y)) = U_y e. x (A ^o (B .o y)))
76	55, 61, 65, 75	syl111anc 1100	. . . . . . 7 |- ((B e. On /\ Lim x) -> (A ^o U_y e. x (B .o y)) = U_y e. x (A ^o (B .o y)))
77	54, 76	eqtrd 1925	. . . . . 6 |- ((B e. On /\ Lim x) -> (A ^o (B .o x)) = U_y e. x (A ^o (B .o y)))
78	51, 77	eqeq12d 1899	. . . . 5 |- ((B e. On /\ Lim x) -> (((A ^o B) ^o x) = (A ^o (B .o x)) <-> U_y e. x ((A ^o B) ^o y) = U_y e. x (A ^o (B .o y))))
79		iuneq2 3273	. . . . 5 |- (A.y e. x ((A ^o B) ^o y) = (A ^o (B .o y)) -> U_y e. x ((A ^o B) ^o y) = U_y e. x (A ^o (B .o y)))
80	78, 79	syl5bir 227	. . . 4 |- ((B e. On /\ Lim x) -> (A.y e. x ((A ^o B) ^o y) = (A ^o (B .o y)) -> ((A ^o B) ^o x) = (A ^o (B .o x))))
81	80	expcom 403	. . 3 |- (Lim x -> (B e. On -> (A.y e. x ((A ^o B) ^o y) = (A ^o (B .o y)) -> ((A ^o B) ^o x) = (A ^o (B .o x)))))
82	4, 8, 12, 16, 27, 41, 81	tfinds3 3948	. 2 |- (C e. On -> (B e. On -> ((A ^o B) ^o C) = (A ^o (B .o C))))
83	82	impcom 378	1 |- ((B e. On /\ C e. On) -> ((A ^o B) ^o C) = (A ^o (B .o C)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  _Vcvv 2292   C_ wss 2593  (/)c0 2875  U_ciun 3255  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659  (class class class)co 4884  1oc1o 5172   +o coa 5174   .o comu 5175   ^o coe 5176

This theorem is referenced by:  oeoe 5274

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-oexp 5181

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