Theorem oeoalem 5271

Description: Lemma for oeoa 5272.

Hypotheses

Ref	Expression
oeoalem.1	|- A e. On
oeoalem.2	|- (/) e. A
oeoalem.3	|- B e. On

Assertion

Ref	Expression
oeoalem	|- (C e. On -> (A ^o (B +o C)) = ((A ^o B) .o (A ^o C)))

Proof of Theorem oeoalem

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . 4 |- (x = (/) -> (B +o x) = (B +o (/)))
2	1	opreq2d 4898	. . 3 |- (x = (/) -> (A ^o (B +o x)) = (A ^o (B +o (/))))
3		opreq2 4890	. . . 4 |- (x = (/) -> (A ^o x) = (A ^o (/)))
4	3	opreq2d 4898	. . 3 |- (x = (/) -> ((A ^o B) .o (A ^o x)) = ((A ^o B) .o (A ^o (/))))
5	2, 4	eqeq12d 1899	. 2 |- (x = (/) -> ((A ^o (B +o x)) = ((A ^o B) .o (A ^o x)) <-> (A ^o (B +o (/))) = ((A ^o B) .o (A ^o (/)))))
6		opreq2 4890	. . . 4 |- (x = y -> (B +o x) = (B +o y))
7	6	opreq2d 4898	. . 3 |- (x = y -> (A ^o (B +o x)) = (A ^o (B +o y)))
8		opreq2 4890	. . . 4 |- (x = y -> (A ^o x) = (A ^o y))
9	8	opreq2d 4898	. . 3 |- (x = y -> ((A ^o B) .o (A ^o x)) = ((A ^o B) .o (A ^o y)))
10	7, 9	eqeq12d 1899	. 2 |- (x = y -> ((A ^o (B +o x)) = ((A ^o B) .o (A ^o x)) <-> (A ^o (B +o y)) = ((A ^o B) .o (A ^o y))))
11		opreq2 4890	. . . 4 |- (x = suc y -> (B +o x) = (B +o suc y))
12	11	opreq2d 4898	. . 3 |- (x = suc y -> (A ^o (B +o x)) = (A ^o (B +o suc y)))
13		opreq2 4890	. . . 4 |- (x = suc y -> (A ^o x) = (A ^o suc y))
14	13	opreq2d 4898	. . 3 |- (x = suc y -> ((A ^o B) .o (A ^o x)) = ((A ^o B) .o (A ^o suc y)))
15	12, 14	eqeq12d 1899	. 2 |- (x = suc y -> ((A ^o (B +o x)) = ((A ^o B) .o (A ^o x)) <-> (A ^o (B +o suc y)) = ((A ^o B) .o (A ^o suc y))))
16		opreq2 4890	. . . 4 |- (x = C -> (B +o x) = (B +o C))
17	16	opreq2d 4898	. . 3 |- (x = C -> (A ^o (B +o x)) = (A ^o (B +o C)))
18		opreq2 4890	. . . 4 |- (x = C -> (A ^o x) = (A ^o C))
19	18	opreq2d 4898	. . 3 |- (x = C -> ((A ^o B) .o (A ^o x)) = ((A ^o B) .o (A ^o C)))
20	17, 19	eqeq12d 1899	. 2 |- (x = C -> ((A ^o (B +o x)) = ((A ^o B) .o (A ^o x)) <-> (A ^o (B +o C)) = ((A ^o B) .o (A ^o C))))
21		oeoalem.1	. . . . 5 |- A e. On
22		oeoalem.3	. . . . 5 |- B e. On
23		oecl 5218	. . . . 5 |- ((A e. On /\ B e. On) -> (A ^o B) e. On)
24	21, 22, 23	mp2an 761	. . . 4 |- (A ^o B) e. On
25		om1 5223	. . . 4 |- ((A ^o B) e. On -> ((A ^o B) .o 1o) = (A ^o B))
26	24, 25	ax-mp 7	. . 3 |- ((A ^o B) .o 1o) = (A ^o B)
27		oe0 5206	. . . . 5 |- (A e. On -> (A ^o (/)) = 1o)
28	21, 27	ax-mp 7	. . . 4 |- (A ^o (/)) = 1o
29	28	opreq2i 4893	. . 3 |- ((A ^o B) .o (A ^o (/))) = ((A ^o B) .o 1o)
30		oa0 5200	. . . . 5 |- (B e. On -> (B +o (/)) = B)
31	22, 30	ax-mp 7	. . . 4 |- (B +o (/)) = B
32	31	opreq2i 4893	. . 3 |- (A ^o (B +o (/))) = (A ^o B)
33	26, 29, 32	3eqtr4ri 1923	. 2 |- (A ^o (B +o (/))) = ((A ^o B) .o (A ^o (/)))
34		oasuc 5208	. . . . . . . 8 |- ((B e. On /\ y e. On) -> (B +o suc y) = suc (B +o y))
35	34	opreq2d 4898	. . . . . . 7 |- ((B e. On /\ y e. On) -> (A ^o (B +o suc y)) = (A ^o suc (B +o y)))
36		oesuc 5211	. . . . . . . 8 |- ((A e. On /\ (B +o y) e. On) -> (A ^o suc (B +o y)) = ((A ^o (B +o y)) .o A))
37		oacl 5215	. . . . . . . 8 |- ((B e. On /\ y e. On) -> (B +o y) e. On)
38	36, 21, 37	sylancr 526	. . . . . . 7 |- ((B e. On /\ y e. On) -> (A ^o suc (B +o y)) = ((A ^o (B +o y)) .o A))
39	35, 38	eqtrd 1925	. . . . . 6 |- ((B e. On /\ y e. On) -> (A ^o (B +o suc y)) = ((A ^o (B +o y)) .o A))
40	22, 39	mpan 759	. . . . 5 |- (y e. On -> (A ^o (B +o suc y)) = ((A ^o (B +o y)) .o A))
41		opreq1 4889	. . . . 5 |- ((A ^o (B +o y)) = ((A ^o B) .o (A ^o y)) -> ((A ^o (B +o y)) .o A) = (((A ^o B) .o (A ^o y)) .o A))
42	40, 41	sylan9eq 1948	. . . 4 |- ((y e. On /\ (A ^o (B +o y)) = ((A ^o B) .o (A ^o y))) -> (A ^o (B +o suc y)) = (((A ^o B) .o (A ^o y)) .o A))
43		oecl 5218	. . . . . . . 8 |- ((A e. On /\ y e. On) -> (A ^o y) e. On)
44		omass 5259	. . . . . . . . 9 |- (((A ^o B) e. On /\ (A ^o y) e. On /\ A e. On) -> (((A ^o B) .o (A ^o y)) .o A) = ((A ^o B) .o ((A ^o y) .o A)))
45	24, 21, 44	mp3an13 1182	. . . . . . . 8 |- ((A ^o y) e. On -> (((A ^o B) .o (A ^o y)) .o A) = ((A ^o B) .o ((A ^o y) .o A)))
46	43, 45	syl 12	. . . . . . 7 |- ((A e. On /\ y e. On) -> (((A ^o B) .o (A ^o y)) .o A) = ((A ^o B) .o ((A ^o y) .o A)))
47		oesuc 5211	. . . . . . . 8 |- ((A e. On /\ y e. On) -> (A ^o suc y) = ((A ^o y) .o A))
48	47	opreq2d 4898	. . . . . . 7 |- ((A e. On /\ y e. On) -> ((A ^o B) .o (A ^o suc y)) = ((A ^o B) .o ((A ^o y) .o A)))
49	46, 48	eqtr4d 1928	. . . . . 6 |- ((A e. On /\ y e. On) -> (((A ^o B) .o (A ^o y)) .o A) = ((A ^o B) .o (A ^o suc y)))
50	21, 49	mpan 759	. . . . 5 |- (y e. On -> (((A ^o B) .o (A ^o y)) .o A) = ((A ^o B) .o (A ^o suc y)))
51	50	adantr 425	. . . 4 |- ((y e. On /\ (A ^o (B +o y)) = ((A ^o B) .o (A ^o y))) -> (((A ^o B) .o (A ^o y)) .o A) = ((A ^o B) .o (A ^o suc y)))
52	42, 51	eqtrd 1925	. . 3 |- ((y e. On /\ (A ^o (B +o y)) = ((A ^o B) .o (A ^o y))) -> (A ^o (B +o suc y)) = ((A ^o B) .o (A ^o suc y)))
53	52	ex 402	. 2 |- (y e. On -> ((A ^o (B +o y)) = ((A ^o B) .o (A ^o y)) -> (A ^o (B +o suc y)) = ((A ^o B) .o (A ^o suc y))))
54		visset 2295	. . . . . . . 8 |- x e. _V
55		oalim 5212	. . . . . . . . 9 |- ((B e. On /\ (x e. _V /\ Lim x)) -> (B +o x) = U_y e. x (B +o y))
56	22, 55	mpan 759	. . . . . . . 8 |- ((x e. _V /\ Lim x) -> (B +o x) = U_y e. x (B +o y))
57	54, 56	mpan 759	. . . . . . 7 |- (Lim x -> (B +o x) = U_y e. x (B +o y))
58	57	opreq2d 4898	. . . . . 6 |- (Lim x -> (A ^o (B +o x)) = (A ^o U_y e. x (B +o y)))
59	54	a1i 8	. . . . . . 7 |- (Lim x -> x e. _V)
60		ordelon 3682	. . . . . . . . . 10 |- ((Ord x /\ y e. x) -> y e. On)
61		limord 3723	. . . . . . . . . 10 |- (Lim x -> Ord x)
62	60, 61	sylan 497	. . . . . . . . 9 |- ((Lim x /\ y e. x) -> y e. On)
63	37, 22, 62	sylancr 526	. . . . . . . 8 |- ((Lim x /\ y e. x) -> (B +o y) e. On)
64	63	r19.21aiva 2176	. . . . . . 7 |- (Lim x -> A.y e. x (B +o y) e. On)
65		0ellim 3726	. . . . . . . 8 |- (Lim x -> (/) e. x)
66		ne0i 2881	. . . . . . . 8 |- ((/) e. x -> x =/= (/))
67	65, 66	syl 12	. . . . . . 7 |- (Lim x -> x =/= (/))
68		visset 2295	. . . . . . . . 9 |- w e. _V
69		oeoalem.2	. . . . . . . . . . 11 |- (/) e. A
70		oelim 5214	. . . . . . . . . . 11 |- (((A e. On /\ (w e. _V /\ Lim w)) /\ (/) e. A) -> (A ^o w) = U_z e. w (A ^o z))
71	69, 70	mpan2 760	. . . . . . . . . 10 |- ((A e. On /\ (w e. _V /\ Lim w)) -> (A ^o w) = U_z e. w (A ^o z))
72	21, 71	mpan 759	. . . . . . . . 9 |- ((w e. _V /\ Lim w) -> (A ^o w) = U_z e. w (A ^o z))
73	68, 72	mpan 759	. . . . . . . 8 |- (Lim w -> (A ^o w) = U_z e. w (A ^o z))
74		oewordi 5266	. . . . . . . . . . 11 |- (((z e. On /\ w e. On /\ A e. On) /\ (/) e. A) -> (z C_ w -> (A ^o z) C_ (A ^o w)))
75	69, 74	mpan2 760	. . . . . . . . . 10 |- ((z e. On /\ w e. On /\ A e. On) -> (z C_ w -> (A ^o z) C_ (A ^o w)))
76	21, 75	mp3an3 1180	. . . . . . . . 9 |- ((z e. On /\ w e. On) -> (z C_ w -> (A ^o z) C_ (A ^o w)))
77	76	3impia 1064	. . . . . . . 8 |- ((z e. On /\ w e. On /\ z C_ w) -> (A ^o z) C_ (A ^o w))
78	73, 77	onopriun 5118	. . . . . . 7 |- ((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)))
79	59, 64, 67, 78	syl111anc 1100	. . . . . 6 |- (Lim x -> (A ^o U_y e. x (B +o y)) = U_y e. x (A ^o (B +o y)))
80	58, 79	eqtrd 1925	. . . . 5 |- (Lim x -> (A ^o (B +o x)) = U_y e. x (A ^o (B +o y)))
81		iuneq2 3273	. . . . 5 |- (A.y e. x (A ^o (B +o y)) = ((A ^o B) .o (A ^o y)) -> U_y e. x (A ^o (B +o y)) = U_y e. x ((A ^o B) .o (A ^o y)))
82	80, 81	sylan9eq 1948	. . . 4 |- ((Lim x /\ A.y e. x (A ^o (B +o y)) = ((A ^o B) .o (A ^o y))) -> (A ^o (B +o x)) = U_y e. x ((A ^o B) .o (A ^o y)))
83		oelim 5214	. . . . . . . . . 10 |- (((A e. On /\ (x e. _V /\ Lim x)) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
84	69, 83	mpan2 760	. . . . . . . . 9 |- ((A e. On /\ (x e. _V /\ Lim x)) -> (A ^o x) = U_y e. x (A ^o y))
85	21, 84	mpan 759	. . . . . . . 8 |- ((x e. _V /\ Lim x) -> (A ^o x) = U_y e. x (A ^o y))
86	54, 85	mpan 759	. . . . . . 7 |- (Lim x -> (A ^o x) = U_y e. x (A ^o y))
87	86	opreq2d 4898	. . . . . 6 |- (Lim x -> ((A ^o B) .o (A ^o x)) = ((A ^o B) .o U_y e. x (A ^o y)))
88	43, 21, 62	sylancr 526	. . . . . . . 8 |- ((Lim x /\ y e. x) -> (A ^o y) e. On)
89	88	r19.21aiva 2176	. . . . . . 7 |- (Lim x -> A.y e. x (A ^o y) e. On)
90		omlim 5213	. . . . . . . . . 10 |- (((A ^o B) e. On /\ (w e. _V /\ Lim w)) -> ((A ^o B) .o w) = U_z e. w ((A ^o B) .o z))
91	24, 90	mpan 759	. . . . . . . . 9 |- ((w e. _V /\ Lim w) -> ((A ^o B) .o w) = U_z e. w ((A ^o B) .o z))
92	68, 91	mpan 759	. . . . . . . 8 |- (Lim w -> ((A ^o B) .o w) = U_z e. w ((A ^o B) .o z))
93		omwordi 5250	. . . . . . . . . 10 |- ((z e. On /\ w e. On /\ (A ^o B) e. On) -> (z C_ w -> ((A ^o B) .o z) C_ ((A ^o B) .o w)))
94	24, 93	mp3an3 1180	. . . . . . . . 9 |- ((z e. On /\ w e. On) -> (z C_ w -> ((A ^o B) .o z) C_ ((A ^o B) .o w)))
95	94	3impia 1064	. . . . . . . 8 |- ((z e. On /\ w e. On /\ z C_ w) -> ((A ^o B) .o z) C_ ((A ^o B) .o w))
96	92, 95	onopriun 5118	. . . . . . 7 |- ((x e. _V /\ A.y e. x (A ^o y) e. On /\ x =/= (/)) -> ((A ^o B) .o U_y e. x (A ^o y)) = U_y e. x ((A ^o B) .o (A ^o y)))
97	59, 89, 67, 96	syl111anc 1100	. . . . . 6 |- (Lim x -> ((A ^o B) .o U_y e. x (A ^o y)) = U_y e. x ((A ^o B) .o (A ^o y)))
98	87, 97	eqtrd 1925	. . . . 5 |- (Lim x -> ((A ^o B) .o (A ^o x)) = U_y e. x ((A ^o B) .o (A ^o y)))
99	98	adantr 425	. . . 4 |- ((Lim x /\ A.y e. x (A ^o (B +o y)) = ((A ^o B) .o (A ^o y))) -> ((A ^o B) .o (A ^o x)) = U_y e. x ((A ^o B) .o (A ^o y)))
100	82, 99	eqtr4d 1928	. . 3 |- ((Lim x /\ A.y e. x (A ^o (B +o y)) = ((A ^o B) .o (A ^o y))) -> (A ^o (B +o x)) = ((A ^o B) .o (A ^o x)))
101	100	ex 402	. 2 |- (Lim x -> (A.y e. x (A ^o (B +o y)) = ((A ^o B) .o (A ^o y)) -> (A ^o (B +o x)) = ((A ^o B) .o (A ^o x))))
102	5, 10, 15, 20, 33, 53, 101	tfinds 3942	1 |- (C e. On -> (A ^o (B +o C)) = ((A ^o B) .o (A ^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:  oeoa 5272

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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