Theorem oev2 4155

Description: Alternate value of ordinal exponentiation. Compare oev 4146.

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 4152	. . . . 5 |- ((/) ^o (/)) = 1o
2		opreq12 3965	. . . . 5 |- ((A = (/) /\ B = (/)) -> (A ^o B) = ((/) ^o (/)))
3		fveq2 3719	. . . . . . . 8 |- (B = (/) -> (rec({<.x, y>. | y = (x .o A)}, 1o)` B) = (rec({<.x, y>. | y = (x .o A)}, 1o)` (/)))
4		1on 4131	. . . . . . . . . 10 |- 1o e. On
5	4	elisseti 1815	. . . . . . . . 9 |- 1o e. V
6	5	rdg0 3936	. . . . . . . 8 |- (rec({<.x, y>. | y = (x .o A)}, 1o)` (/)) = 1o
7	3, 6	syl6eq 1521	. . . . . . 7 |- (B = (/) -> (rec({<.x, y>. | y = (x .o A)}, 1o)` B) = 1o)
8		inteq 2532	. . . . . . . 8 |- (B = (/) -> |^|B = |^|(/))
9		int0 2543	. . . . . . . 8 |- |^|(/) = V
10	8, 9	syl6eq 1521	. . . . . . 7 |- (B = (/) -> |^|B = V)
11	7, 10	ineq12d 2215	. . . . . 6 |- (B = (/) -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B) = (1o i^i V))
12		inv1 2296	. . . . . . 7 |- (1o i^i V) = 1o
13	12	a1i 8	. . . . . 6 |- (A = (/) -> (1o i^i V) = 1o)
14	11, 13	sylan9eqr 1527	. . . . 5 |- ((A = (/) /\ B = (/)) -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B) = 1o)
15	1, 2, 14	3eqtr4a 1530	. . . 4 |- ((A = (/) /\ B = (/)) -> (A ^o B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B))
16		opreq1 3963	. . . . . . 7 |- (A = (/) -> (A ^o B) = ((/) ^o B))
17		oe0m1 4153	. . . . . . . 8 |- (B e. On -> ((/) e. B <-> ((/) ^o B) = (/)))
18	17	biimpa 416	. . . . . . 7 |- ((B e. On /\ (/) e. B) -> ((/) ^o B) = (/))
19	16, 18	sylan9eqr 1527	. . . . . 6 |- (((B e. On /\ (/) e. B) /\ A = (/)) -> (A ^o B) = (/))
20	19	an1rs 489	. . . . 5 |- (((B e. On /\ A = (/)) /\ (/) e. B) -> (A ^o B) = (/))
21		int0el 2557	. . . . . . . 8 |- ((/) e. B -> |^|B = (/))
22	21	ineq2d 2214	. . . . . . 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 2295	. . . . . . 7 |- ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i (/)) = (/)
24	22, 23	syl6eq 1521	. . . . . 6 |- ((/) e. B -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B) = (/))
25	24	adantl 388	. . . . 5 |- (((B e. On /\ A = (/)) /\ (/) e. B) -> ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B) = (/))
26	20, 25	eqtr4d 1508	. . . 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 4145	. . 3 |- ((B e. On /\ A = (/)) -> (A ^o B) = ((rec({<.x, y>. | y = (x .o A)}, 1o)` B) i^i |^|B))
28		inteq 2532	. . . . . . . . . 10 |- (A = (/) -> |^|A = |^|(/))
29	28, 9	syl6eq 1521	. . . . . . . . 9 |- (A = (/) -> |^|A = V)
30	29	difeq2d 2156	. . . . . . . 8 |- (A = (/) -> (V \ |^|A) = (V \ V))
31		difid 2331	. . . . . . . 8 |- (V \ V) = (/)
32	30, 31	syl6eq 1521	. . . . . . 7 |- (A = (/) -> (V \ |^|A) = (/))
33	32	uneq2d 2181	. . . . . 6 |- (A = (/) -> (|^|B u. (V \ |^|A)) = (|^|B u. (/)))
34		uncom 2173	. . . . . 6 |- (|^|B u. (V \ |^|A)) = ((V \ |^|A) u. |^|B)
35		un0 2294	. . . . . 6 |- (|^|B u. (/)) = |^|B
36	33, 34, 35	3eqtr3g 1528	. . . . 5 |- (A = (/) -> ((V \ |^|A) u. |^|B) = |^|B)
37	36	adantl 388	. . . 4 |- ((B e. On /\ A = (/)) -> ((V \ |^|A) u. |^|B) = |^|B)
38	37	ineq2d 2214	. . 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 1508	. 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 4147	. . 3 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) = (rec({<.x, y>. | y = (x .o A)}, 1o)` B))
41		int0el 2557	. . . . . . . . . 10 |- ((/) e. A -> |^|A = (/))
42	41	difeq2d 2156	. . . . . . . . 9 |- ((/) e. A -> (V \ |^|A) = (V \ (/)))
43		dif0 2332	. . . . . . . . 9 |- (V \ (/)) = V
44	42, 43	syl6eq 1521	. . . . . . . 8 |- ((/) e. A -> (V \ |^|A) = V)
45	44	uneq2d 2181	. . . . . . 7 |- ((/) e. A -> (|^|B u. (V \ |^|A)) = (|^|B u. V))
46		unv 2297	. . . . . . 7 |- (|^|B u. V) = V
47	45, 34, 46	3eqtr3g 1528	. . . . . 6 |- ((/) e. A -> ((V \ |^|A) u. |^|B) = V)
48	47	adantl 388	. . . . 5 |- (((A e. On /\ B e. On) /\ (/) e. A) -> ((V \ |^|A) u. |^|B) = V)
49	48	ineq2d 2214	. . . 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 2296	. . . 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 1522	. . 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 1505	. 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 4145	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 223   = wceq 955   e. wcel 957  Vcvv 1808   \ cdif 2041   u. cun 2042   i^i cin 2043  (/)c0 2277  |^|cint 2529  {copab 2662  Oncon0 2944  ` cfv 3178  reccrdg 3926  (class class class)co 3958  1oc1o 4121   .o comu 4124   ^o coe 4125

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1o 4126  df-oexp 4130

Copyright terms: Public domain