Theorem oesuclem 7605

Description: Lemma for oesuc 7607. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypotheses

Ref	Expression
oesuclem.1	|- Lim X
oesuclem.2	|- ( B e. X -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) )

Assertion

Ref	Expression
oesuclem	|- ( ( A e. On /\ B e. X ) -> ( A ^o suc B ) = ( ( A ^o B ) .o A ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   X( x)

Proof of Theorem oesuclem

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( A = (/) -> ( A ^o suc B ) = ( (/) ^o suc B ) )
2		oesuclem.1	. . . . . . . 8 |- Lim X
3		limord 5784	. . . . . . . 8 |- ( Lim X -> Ord X )
4	2, 3	ax-mp 5	. . . . . . 7 |- Ord X
5		ordelord 5745	. . . . . . 7 |- ( ( Ord X /\ B e. X ) -> Ord B )
6	4, 5	mpan 706	. . . . . 6 |- ( B e. X -> Ord B )
7		0elsuc 7035	. . . . . 6 |- ( Ord B -> (/) e. suc B )
8	6, 7	syl 17	. . . . 5 |- ( B e. X -> (/) e. suc B )
9		limsuc 7049	. . . . . . 7 |- ( Lim X -> ( B e. X <-> suc B e. X ) )
10	2, 9	ax-mp 5	. . . . . 6 |- ( B e. X <-> suc B e. X )
11		ordelon 5747	. . . . . . . 8 |- ( ( Ord X /\ suc B e. X ) -> suc B e. On )
12	4, 11	mpan 706	. . . . . . 7 |- ( suc B e. X -> suc B e. On )
13		oe0m1 7601	. . . . . . 7 |- ( suc B e. On -> ( (/) e. suc B <-> ( (/) ^o suc B ) = (/) ) )
14	12, 13	syl 17	. . . . . 6 |- ( suc B e. X -> ( (/) e. suc B <-> ( (/) ^o suc B ) = (/) ) )
15	10, 14	sylbi 207	. . . . 5 |- ( B e. X -> ( (/) e. suc B <-> ( (/) ^o suc B ) = (/) ) )
16	8, 15	mpbid 222	. . . 4 |- ( B e. X -> ( (/) ^o suc B ) = (/) )
17	1, 16	sylan9eqr 2678	. . 3 |- ( ( B e. X /\ A = (/) ) -> ( A ^o suc B ) = (/) )
18		oveq1 6657	. . . . 5 |- ( A = (/) -> ( A ^o B ) = ( (/) ^o B ) )
19		id 22	. . . . 5 |- ( A = (/) -> A = (/) )
20	18, 19	oveq12d 6668	. . . 4 |- ( A = (/) -> ( ( A ^o B ) .o A ) = ( ( (/) ^o B ) .o (/) ) )
21		ordelon 5747	. . . . . . 7 |- ( ( Ord X /\ B e. X ) -> B e. On )
22	4, 21	mpan 706	. . . . . 6 |- ( B e. X -> B e. On )
23		oveq2 6658	. . . . . . . . 9 |- ( B = (/) -> ( (/) ^o B ) = ( (/) ^o (/) ) )
24		oe0m0 7600	. . . . . . . . . 10 |- ( (/) ^o (/) ) = 1o
25		1on 7567	. . . . . . . . . 10 |- 1o e. On
26	24, 25	eqeltri 2697	. . . . . . . . 9 |- ( (/) ^o (/) ) e. On
27	23, 26	syl6eqel 2709	. . . . . . . 8 |- ( B = (/) -> ( (/) ^o B ) e. On )
28	27	adantl 482	. . . . . . 7 |- ( ( B e. X /\ B = (/) ) -> ( (/) ^o B ) e. On )
29		oe0m1 7601	. . . . . . . . . . 11 |- ( B e. On -> ( (/) e. B <-> ( (/) ^o B ) = (/) ) )
30	22, 29	syl 17	. . . . . . . . . 10 |- ( B e. X -> ( (/) e. B <-> ( (/) ^o B ) = (/) ) )
31	30	biimpa 501	. . . . . . . . 9 |- ( ( B e. X /\ (/) e. B ) -> ( (/) ^o B ) = (/) )
32		0elon 5778	. . . . . . . . 9 |- (/) e. On
33	31, 32	syl6eqel 2709	. . . . . . . 8 |- ( ( B e. X /\ (/) e. B ) -> ( (/) ^o B ) e. On )
34	33	adantll 750	. . . . . . 7 |- ( ( ( B e. On /\ B e. X ) /\ (/) e. B ) -> ( (/) ^o B ) e. On )
35	28, 34	oe0lem 7593	. . . . . 6 |- ( ( B e. On /\ B e. X ) -> ( (/) ^o B ) e. On )
36	22, 35	mpancom 703	. . . . 5 |- ( B e. X -> ( (/) ^o B ) e. On )
37		om0 7597	. . . . 5 |- ( ( (/) ^o B ) e. On -> ( ( (/) ^o B ) .o (/) ) = (/) )
38	36, 37	syl 17	. . . 4 |- ( B e. X -> ( ( (/) ^o B ) .o (/) ) = (/) )
39	20, 38	sylan9eqr 2678	. . 3 |- ( ( B e. X /\ A = (/) ) -> ( ( A ^o B ) .o A ) = (/) )
40	17, 39	eqtr4d 2659	. 2 |- ( ( B e. X /\ A = (/) ) -> ( A ^o suc B ) = ( ( A ^o B ) .o A ) )
41		oesuclem.2	. . . 4 |- ( B e. X -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) )
42	41	ad2antlr 763	. . 3 |- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) )
43	10, 12	sylbi 207	. . . 4 |- ( B e. X -> suc B e. On )
44		oevn0 7595	. . . 4 |- ( ( ( A e. On /\ suc B e. On ) /\ (/) e. A ) -> ( A ^o suc B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) )
45	43, 44	sylanl2 683	. . 3 |- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( A ^o suc B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` suc B ) )
46		ovex 6678	. . . . 5 |- ( A ^o B ) e. _V
47		oveq1 6657	. . . . . 6 |- ( x = ( A ^o B ) -> ( x .o A ) = ( ( A ^o B ) .o A ) )
48		eqid 2622	. . . . . 6 |- ( x e. _V |-> ( x .o A ) ) = ( x e. _V |-> ( x .o A ) )
49		ovex 6678	. . . . . 6 |- ( ( A ^o B ) .o A ) e. _V
50	47, 48, 49	fvmpt 6282	. . . . 5 |- ( ( A ^o B ) e. _V -> ( ( x e. _V |-> ( x .o A ) ) ` ( A ^o B ) ) = ( ( A ^o B ) .o A ) )
51	46, 50	ax-mp 5	. . . 4 |- ( ( x e. _V |-> ( x .o A ) ) ` ( A ^o B ) ) = ( ( A ^o B ) .o A )
52		oevn0 7595	. . . . . 6 |- ( ( ( A e. On /\ B e. On ) /\ (/) e. A ) -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )
53	22, 52	sylanl2 683	. . . . 5 |- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( A ^o B ) = ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) )
54	53	fveq2d 6195	. . . 4 |- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( ( x e. _V |-> ( x .o A ) ) ` ( A ^o B ) ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) )
55	51, 54	syl5eqr 2670	. . 3 |- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( ( A ^o B ) .o A ) = ( ( x e. _V |-> ( x .o A ) ) ` ( rec ( ( x e. _V |-> ( x .o A ) ) , 1o ) ` B ) ) )
56	42, 45, 55	3eqtr4d 2666	. 2 |- ( ( ( A e. On /\ B e. X ) /\ (/) e. A ) -> ( A ^o suc B ) = ( ( A ^o B ) .o A ) )
57	40, 56	oe0lem 7593	1 |- ( ( A e. On /\ B e. X ) -> ( A ^o suc B ) = ( ( A ^o B ) .o A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   ` cfv 5888  (class class class)co 6650   reccrdg 7505   1oc1o 7553   .o comu 7558   ^o coe 7559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-omul 7565  df-oexp 7566

This theorem is referenced by:  oesuc  7607  onesuc  7610