Theorem oev2 5207

Description: Alternate value of ordinal exponentiation. Compare oev 5198.

Assertion

Ref	Expression
oev2	|- ((A e. On /\ B e. On) -> (A ^o B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i ((_V \ |^|A) u. |^|B)))

Distinct variable group:   x,y,A

Proof of Theorem oev2

Step	Hyp	Ref	Expression
1		oe0m0 5204	. . . . 5 |- ((/) ^o (/)) = 1o
2		opreq12 4891	. . . . 5 |- ((A = (/) /\ B = (/)) -> (A ^o B) = ((/) ^o (/)))
3		fveq2 4681	. . . . . . . 8 |- (B = (/) -> (rec({<.x, y>. | y = (x .o A)}, 1o)` B) = (rec({<.x, y>. | y = (x .o A)}, 1o)` (/)))
4		1on 5182	. . . . . . . . . 10 |- 1o e. On
5	4	elisseti 2301	. . . . . . . . 9 |- 1o e. _V
6	5	rdg0 5149	. . . . . . . 8 |- (rec({<.x, y>. | y = (x .o A)}, 1o)` (/)) = 1o
7	3, 6	syl6eq 1944	. . . . . . 7 |- (B = (/) -> (rec({<.x, y>. | y = (x .o A)}, 1o)` B) = 1o)
8		inteq 3217	. . . . . . . 8 |- (B = (/) -> |^|B = |^|(/))
9		int0 3230	. . . . . . . 8 |- |^|(/) = _V
10	8, 9	syl6eq 1944	. . . . . . 7 |- (B = (/) -> |^|B = _V)
11	7, 10	ineq12d 2797	. . . . . 6 |- (B = (/) -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B) = (1o i^i _V))
12		inv1 2898	. . . . . . 7 |- (1o i^i _V) = 1o
13	12	a1i 8	. . . . . 6 |- (A = (/) -> (1o i^i _V) = 1o)
14	11, 13	sylan9eqr 1951	. . . . 5 |- ((A = (/) /\ B = (/)) -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B) = 1o)
15	1, 2, 14	3eqtr4a 1954	. . . 4 |- ((A = (/) /\ B = (/)) -> (A ^o B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B))
16		opreq1 4889	. . . . . . 7 |- (A = (/) -> (A ^o B) = ((/) ^o B))
17		oe0m1 5205	. . . . . . . 8 |- (B e. On -> ((/) e. B <-> ((/) ^o B) = (/)))
18	17	biimpa 460	. . . . . . 7 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) = (/))
19	16, 18	sylan9eqr 1951	. . . . . 6 |- (((B e. On /\ (/) e. B) /\ A = (/)) -> (A ^o B) = (/))
20	19	an1rs 547	. . . . 5 |- (((B e. On /\ A = (/)) /\ (/) e. B) -> (A ^o B) = (/))
21		int0el 3248	. . . . . . . 8 |- ((/) e. B -> |^|B = (/))
22	21	ineq2d 2796	. . . . . . 7 |- ((/) e. B -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i (/)))
23		in0 2897	. . . . . . 7 |- ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i (/)) = (/)
24	22, 23	syl6eq 1944	. . . . . 6 |- ((/) e. B -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B) = (/))
25	24	adantl 424	. . . . 5 |- (((B e. On /\ A = (/)) /\ (/) e. B) -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B) = (/))
26	20, 25	eqtr4d 1928	. . . 4 |- (((B e. On /\ A = (/)) /\ (/) e. B) -> (A ^o B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B))
27	15, 26	oe0lem 5197	. . 3 |- ((B e. On /\ A = (/)) -> (A ^o B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B))
28		inteq 3217	. . . . . . . . . 10 |- (A = (/) -> |^|A = |^|(/))
29	28, 9	syl6eq 1944	. . . . . . . . 9 |- (A = (/) -> |^|A = _V)
30	29	difeq2d 2726	. . . . . . . 8 |- (A = (/) -> (_V \ |^|A) = (_V \ _V))
31		difid 2942	. . . . . . . 8 |- (_V \ _V) = (/)
32	30, 31	syl6eq 1944	. . . . . . 7 |- (A = (/) -> (_V \ |^|A) = (/))
33	32	uneq2d 2755	. . . . . 6 |- (A = (/) -> (|^|B u. (_V \ |^|A)) = (|^|B u. (/)))
34		uncom 2744	. . . . . 6 |- (|^|B u. (_V \ |^|A)) = ((_V \ |^|A) u. |^|B)
35		un0 2896	. . . . . 6 |- (|^|B u. (/)) = |^|B
36	33, 34, 35	3eqtr3g 1952	. . . . 5 |- (A = (/) -> ((_V \ |^|A) u. |^|B) = |^|B)
37	36	adantl 424	. . . 4 |- ((B e. On /\ A = (/)) -> ((_V \ |^|A) u. |^|B) = |^|B)
38	37	ineq2d 2796	. . 3 |- ((B e. On /\ A = (/)) -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i ((_V \ |^|A) u. |^|B)) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B))
39	27, 38	eqtr4d 1928	. 2 |- ((B e. On /\ A = (/)) -> (A ^o B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i ((_V \ |^|A) u. |^|B)))
40		oevn0 5199	. . 3 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) = (rec({<.x, y>. | y = (x .o A)}, 1o)` B))
41		int0el 3248	. . . . . . . . . 10 |- ((/) e. A -> |^|A = (/))
42	41	difeq2d 2726	. . . . . . . . 9 |- ((/) e. A -> (_V \ |^|A) = (_V \ (/)))
43		dif0 2943	. . . . . . . . 9 |- (_V \ (/)) = _V
44	42, 43	syl6eq 1944	. . . . . . . 8 |- ((/) e. A -> (_V \ |^|A) = _V)
45	44	uneq2d 2755	. . . . . . 7 |- ((/) e. A -> (|^|B u. (_V \ |^|A)) = (|^|B u. _V))
46		unv 2899	. . . . . . 7 |- (|^|B u. _V) = _V
47	45, 34, 46	3eqtr3g 1952	. . . . . 6 |- ((/) e. A -> ((_V \ |^|A) u. |^|B) = _V)
48	47	adantl 424	. . . . 5 |- (((A e. On /\ B e. On) /\ (/) e. A) -> ((_V \ |^|A) u. |^|B) = _V)
49	48	ineq2d 2796	. . . 4 |- (((A e. On /\ B e. On) /\ (/) e. A) -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i ((_V \ |^|A) u. |^|B)) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i _V))
50		inv1 2898	. . . 4 |- ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i _V) = (rec({<.x, y>. | y = (x .o A)}, 1o)` B)
51	49, 50	syl6req 1945	. . 3 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (rec({<.x, y>. | y = (x .o A)}, 1o)` B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i ((_V \ |^|A) u. |^|B)))
52	40, 51	eqtrd 1925	. 2 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i ((_V \ |^|A) u. |^|B)))
53	39, 52	oe0lem 5197	1 |- ((A e. On /\ B e. On) -> (A ^o B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i ((_V \ |^|A) u. |^|B)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592  (/)c0 2875  |^|cint 3214  {copab 3395  Oncon0 3657  ` cfv 3998  (class class class)co 4884  reccrdg 5139  1oc1o 5172   .o comu 5175   ^o coe 5176

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-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-oexp 5181

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