Theorem rdgeqoa 33218

Description: If a recursive function with an initial value A at step N is equal to itself with an initial value B at step M, then every finite number of successor steps will also be equal. (Contributed by ML, 21-Oct-2020.)

Assertion

Ref	Expression
rdgeqoa	|- ( ( N e. On /\ M e. On /\ X e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o X ) ) = ( rec ( F , B ) ` ( M +o X ) ) ) )

Proof of Theorem rdgeqoa

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1063	. 2 |- ( ( N e. On /\ M e. On /\ X e. om ) -> X e. om )
2		eleq1 2689	. . . . 5 |- ( x = X -> ( x e. om <-> X e. om ) )
3	2	3anbi3d 1405	. . . 4 |- ( x = X -> ( ( N e. On /\ M e. On /\ x e. om ) <-> ( N e. On /\ M e. On /\ X e. om ) ) )
4		oveq2 6658	. . . . . . 7 |- ( x = X -> ( N +o x ) = ( N +o X ) )
5	4	fveq2d 6195	. . . . . 6 |- ( x = X -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , A ) ` ( N +o X ) ) )
6		oveq2 6658	. . . . . . 7 |- ( x = X -> ( M +o x ) = ( M +o X ) )
7	6	fveq2d 6195	. . . . . 6 |- ( x = X -> ( rec ( F , B ) ` ( M +o x ) ) = ( rec ( F , B ) ` ( M +o X ) ) )
8	5, 7	eqeq12d 2637	. . . . 5 |- ( x = X -> ( ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) <-> ( rec ( F , A ) ` ( N +o X ) ) = ( rec ( F , B ) ` ( M +o X ) ) ) )
9	8	imbi2d 330	. . . 4 |- ( x = X -> ( ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) <-> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o X ) ) = ( rec ( F , B ) ` ( M +o X ) ) ) ) )
10	3, 9	imbi12d 334	. . 3 |- ( x = X -> ( ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) <-> ( ( N e. On /\ M e. On /\ X e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o X ) ) = ( rec ( F , B ) ` ( M +o X ) ) ) ) ) )
11		peano1 7085	. . . . 5 |- (/) e. om
12		oa0 7596	. . . . . . . . . . . 12 |- ( N e. On -> ( N +o (/) ) = N )
13	12	fveq2d 6195	. . . . . . . . . . 11 |- ( N e. On -> ( rec ( F , A ) ` ( N +o (/) ) ) = ( rec ( F , A ) ` N ) )
14	13	eqcomd 2628	. . . . . . . . . 10 |- ( N e. On -> ( rec ( F , A ) ` N ) = ( rec ( F , A ) ` ( N +o (/) ) ) )
15		oa0 7596	. . . . . . . . . . . 12 |- ( M e. On -> ( M +o (/) ) = M )
16	15	fveq2d 6195	. . . . . . . . . . 11 |- ( M e. On -> ( rec ( F , B ) ` ( M +o (/) ) ) = ( rec ( F , B ) ` M ) )
17	16	eqcomd 2628	. . . . . . . . . 10 |- ( M e. On -> ( rec ( F , B ) ` M ) = ( rec ( F , B ) ` ( M +o (/) ) ) )
18	14, 17	eqeqan12d 2638	. . . . . . . . 9 |- ( ( N e. On /\ M e. On ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) <-> ( rec ( F , A ) ` ( N +o (/) ) ) = ( rec ( F , B ) ` ( M +o (/) ) ) ) )
19	18	biimpd 219	. . . . . . . 8 |- ( ( N e. On /\ M e. On ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o (/) ) ) = ( rec ( F , B ) ` ( M +o (/) ) ) ) )
20		eleq1 2689	. . . . . . . . . . 11 |- ( x = (/) -> ( x e. om <-> (/) e. om ) )
21	20	3anbi3d 1405	. . . . . . . . . 10 |- ( x = (/) -> ( ( N e. On /\ M e. On /\ x e. om ) <-> ( N e. On /\ M e. On /\ (/) e. om ) ) )
22	11	biantru 526	. . . . . . . . . . . 12 |- ( M e. On <-> ( M e. On /\ (/) e. om ) )
23	22	anbi2i 730	. . . . . . . . . . 11 |- ( ( N e. On /\ M e. On ) <-> ( N e. On /\ ( M e. On /\ (/) e. om ) ) )
24		3anass 1042	. . . . . . . . . . 11 |- ( ( N e. On /\ M e. On /\ (/) e. om ) <-> ( N e. On /\ ( M e. On /\ (/) e. om ) ) )
25	23, 24	bitr4i 267	. . . . . . . . . 10 |- ( ( N e. On /\ M e. On ) <-> ( N e. On /\ M e. On /\ (/) e. om ) )
26	21, 25	syl6bbr 278	. . . . . . . . 9 |- ( x = (/) -> ( ( N e. On /\ M e. On /\ x e. om ) <-> ( N e. On /\ M e. On ) ) )
27		oveq2 6658	. . . . . . . . . . . 12 |- ( x = (/) -> ( N +o x ) = ( N +o (/) ) )
28	27	fveq2d 6195	. . . . . . . . . . 11 |- ( x = (/) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , A ) ` ( N +o (/) ) ) )
29		oveq2 6658	. . . . . . . . . . . 12 |- ( x = (/) -> ( M +o x ) = ( M +o (/) ) )
30	29	fveq2d 6195	. . . . . . . . . . 11 |- ( x = (/) -> ( rec ( F , B ) ` ( M +o x ) ) = ( rec ( F , B ) ` ( M +o (/) ) ) )
31	28, 30	eqeq12d 2637	. . . . . . . . . 10 |- ( x = (/) -> ( ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) <-> ( rec ( F , A ) ` ( N +o (/) ) ) = ( rec ( F , B ) ` ( M +o (/) ) ) ) )
32	31	imbi2d 330	. . . . . . . . 9 |- ( x = (/) -> ( ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) <-> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o (/) ) ) = ( rec ( F , B ) ` ( M +o (/) ) ) ) ) )
33	26, 32	imbi12d 334	. . . . . . . 8 |- ( x = (/) -> ( ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) <-> ( ( N e. On /\ M e. On ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o (/) ) ) = ( rec ( F , B ) ` ( M +o (/) ) ) ) ) ) )
34	19, 33	mpbiri 248	. . . . . . 7 |- ( x = (/) -> ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) )
35	34	ax-gen 1722	. . . . . 6 |- A. x ( x = (/) -> ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) )
36		sbc6g 3461	. . . . . 6 |- ( (/) e. om -> ( [. (/) / x ]. ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) <-> A. x ( x = (/) -> ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) ) ) )
37	35, 36	mpbiri 248	. . . . 5 |- ( (/) e. om -> [. (/) / x ]. ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) )
38	11, 37	ax-mp 5	. . . 4 |- [. (/) / x ]. ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) )
39		peano2b 7081	. . . . 5 |- ( x e. om <-> suc x e. om )
40	39	3anbi3i 1255	. . . . . . . 8 |- ( ( N e. On /\ M e. On /\ x e. om ) <-> ( N e. On /\ M e. On /\ suc x e. om ) )
41	40	imbi1i 339	. . . . . . 7 |- ( ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) <-> ( ( N e. On /\ M e. On /\ suc x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) )
42		nnon 7071	. . . . . . . . . . . . 13 |- ( x e. om -> x e. On )
43		oacl 7615	. . . . . . . . . . . . . . . . 17 |- ( ( N e. On /\ x e. On ) -> ( N +o x ) e. On )
44		oacl 7615	. . . . . . . . . . . . . . . . 17 |- ( ( M e. On /\ x e. On ) -> ( M +o x ) e. On )
45	43, 44	anim12i 590	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. On /\ x e. On ) /\ ( M e. On /\ x e. On ) ) -> ( ( N +o x ) e. On /\ ( M +o x ) e. On ) )
46	45	3impdir 1382	. . . . . . . . . . . . . . 15 |- ( ( N e. On /\ M e. On /\ x e. On ) -> ( ( N +o x ) e. On /\ ( M +o x ) e. On ) )
47		rdgsuc 7520	. . . . . . . . . . . . . . . . . 18 |- ( ( N +o x ) e. On -> ( rec ( F , A ) ` suc ( N +o x ) ) = ( F ` ( rec ( F , A ) ` ( N +o x ) ) ) )
48		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) -> ( F ` ( rec ( F , A ) ` ( N +o x ) ) ) = ( F ` ( rec ( F , B ) ` ( M +o x ) ) ) )
49	47, 48	sylan9eqr 2678	. . . . . . . . . . . . . . . . 17 |- ( ( ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) /\ ( N +o x ) e. On ) -> ( rec ( F , A ) ` suc ( N +o x ) ) = ( F ` ( rec ( F , B ) ` ( M +o x ) ) ) )
50	49	adantrr 753	. . . . . . . . . . . . . . . 16 |- ( ( ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) /\ ( ( N +o x ) e. On /\ ( M +o x ) e. On ) ) -> ( rec ( F , A ) ` suc ( N +o x ) ) = ( F ` ( rec ( F , B ) ` ( M +o x ) ) ) )
51		rdgsuc 7520	. . . . . . . . . . . . . . . . 17 |- ( ( M +o x ) e. On -> ( rec ( F , B ) ` suc ( M +o x ) ) = ( F ` ( rec ( F , B ) ` ( M +o x ) ) ) )
52	51	ad2antll 765	. . . . . . . . . . . . . . . 16 |- ( ( ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) /\ ( ( N +o x ) e. On /\ ( M +o x ) e. On ) ) -> ( rec ( F , B ) ` suc ( M +o x ) ) = ( F ` ( rec ( F , B ) ` ( M +o x ) ) ) )
53	50, 52	eqtr4d 2659	. . . . . . . . . . . . . . 15 |- ( ( ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) /\ ( ( N +o x ) e. On /\ ( M +o x ) e. On ) ) -> ( rec ( F , A ) ` suc ( N +o x ) ) = ( rec ( F , B ) ` suc ( M +o x ) ) )
54	46, 53	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) /\ ( N e. On /\ M e. On /\ x e. On ) ) -> ( rec ( F , A ) ` suc ( N +o x ) ) = ( rec ( F , B ) ` suc ( M +o x ) ) )
55	54	ancoms 469	. . . . . . . . . . . . 13 |- ( ( ( N e. On /\ M e. On /\ x e. On ) /\ ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) -> ( rec ( F , A ) ` suc ( N +o x ) ) = ( rec ( F , B ) ` suc ( M +o x ) ) )
56	42, 55	syl3anl3 1376	. . . . . . . . . . . 12 |- ( ( ( N e. On /\ M e. On /\ x e. om ) /\ ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) -> ( rec ( F , A ) ` suc ( N +o x ) ) = ( rec ( F , B ) ` suc ( M +o x ) ) )
57		onasuc 7608	. . . . . . . . . . . . . . 15 |- ( ( N e. On /\ x e. om ) -> ( N +o suc x ) = suc ( N +o x ) )
58	57	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( N e. On /\ x e. om ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , A ) ` suc ( N +o x ) ) )
59	58	3adant2 1080	. . . . . . . . . . . . 13 |- ( ( N e. On /\ M e. On /\ x e. om ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , A ) ` suc ( N +o x ) ) )
60	59	adantr 481	. . . . . . . . . . . 12 |- ( ( ( N e. On /\ M e. On /\ x e. om ) /\ ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , A ) ` suc ( N +o x ) ) )
61		onasuc 7608	. . . . . . . . . . . . . . 15 |- ( ( M e. On /\ x e. om ) -> ( M +o suc x ) = suc ( M +o x ) )
62	61	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( M e. On /\ x e. om ) -> ( rec ( F , B ) ` ( M +o suc x ) ) = ( rec ( F , B ) ` suc ( M +o x ) ) )
63	62	3adant1 1079	. . . . . . . . . . . . 13 |- ( ( N e. On /\ M e. On /\ x e. om ) -> ( rec ( F , B ) ` ( M +o suc x ) ) = ( rec ( F , B ) ` suc ( M +o x ) ) )
64	63	adantr 481	. . . . . . . . . . . 12 |- ( ( ( N e. On /\ M e. On /\ x e. om ) /\ ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) -> ( rec ( F , B ) ` ( M +o suc x ) ) = ( rec ( F , B ) ` suc ( M +o x ) ) )
65	56, 60, 64	3eqtr4d 2666	. . . . . . . . . . 11 |- ( ( ( N e. On /\ M e. On /\ x e. om ) /\ ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) )
66	65	ex 450	. . . . . . . . . 10 |- ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) )
67	66	imim2d 57	. . . . . . . . 9 |- ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) ) )
68	40, 67	sylbir 225	. . . . . . . 8 |- ( ( N e. On /\ M e. On /\ suc x e. om ) -> ( ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) ) )
69	68	a2i 14	. . . . . . 7 |- ( ( ( N e. On /\ M e. On /\ suc x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) -> ( ( N e. On /\ M e. On /\ suc x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) ) )
70	41, 69	sylbi 207	. . . . . 6 |- ( ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) -> ( ( N e. On /\ M e. On /\ suc x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) ) )
71		sbcimg 3477	. . . . . . 7 |- ( suc x e. om -> ( [. suc x / x ]. ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) <-> ( [. suc x / x ]. ( N e. On /\ M e. On /\ x e. om ) -> [. suc x / x ]. ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) ) )
72		sbc3an 3494	. . . . . . . . 9 |- ( [. suc x / x ]. ( N e. On /\ M e. On /\ x e. om ) <-> ( [. suc x / x ]. N e. On /\ [. suc x / x ]. M e. On /\ [. suc x / x ]. x e. om ) )
73		sbcg 3503	. . . . . . . . . 10 |- ( suc x e. om -> ( [. suc x / x ]. N e. On <-> N e. On ) )
74		sbcg 3503	. . . . . . . . . 10 |- ( suc x e. om -> ( [. suc x / x ]. M e. On <-> M e. On ) )
75		sbcel1v 3495	. . . . . . . . . . 11 |- ( [. suc x / x ]. x e. om <-> suc x e. om )
76	75	a1i 11	. . . . . . . . . 10 |- ( suc x e. om -> ( [. suc x / x ]. x e. om <-> suc x e. om ) )
77	73, 74, 76	3anbi123d 1399	. . . . . . . . 9 |- ( suc x e. om -> ( ( [. suc x / x ]. N e. On /\ [. suc x / x ]. M e. On /\ [. suc x / x ]. x e. om ) <-> ( N e. On /\ M e. On /\ suc x e. om ) ) )
78	72, 77	syl5bb 272	. . . . . . . 8 |- ( suc x e. om -> ( [. suc x / x ]. ( N e. On /\ M e. On /\ x e. om ) <-> ( N e. On /\ M e. On /\ suc x e. om ) ) )
79		sbcimg 3477	. . . . . . . . 9 |- ( suc x e. om -> ( [. suc x / x ]. ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) <-> ( [. suc x / x ]. ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> [. suc x / x ]. ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) )
80		sbcg 3503	. . . . . . . . . 10 |- ( suc x e. om -> ( [. suc x / x ]. ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) <-> ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) ) )
81		sbceqg 3984	. . . . . . . . . . 11 |- ( suc x e. om -> ( [. suc x / x ]. ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) <-> [_ suc x / x ]_ ( rec ( F , A ) ` ( N +o x ) ) = [_ suc x / x ]_ ( rec ( F , B ) ` ( M +o x ) ) ) )
82		csbfv12 6231	. . . . . . . . . . . . 13 |- [_ suc x / x ]_ ( rec ( F , A ) ` ( N +o x ) ) = ( [_ suc x / x ]_ rec ( F , A ) ` [_ suc x / x ]_ ( N +o x ) )
83		csbconstg 3546	. . . . . . . . . . . . . 14 |- ( suc x e. om -> [_ suc x / x ]_ rec ( F , A ) = rec ( F , A ) )
84		csbov123 6687	. . . . . . . . . . . . . . 15 |- [_ suc x / x ]_ ( N +o x ) = ( [_ suc x / x ]_ N [_ suc x / x ]_ +o [_ suc x / x ]_ x )
85		csbconstg 3546	. . . . . . . . . . . . . . . 16 |- ( suc x e. om -> [_ suc x / x ]_ +o = +o )
86		csbconstg 3546	. . . . . . . . . . . . . . . 16 |- ( suc x e. om -> [_ suc x / x ]_ N = N )
87		csbvarg 4003	. . . . . . . . . . . . . . . 16 |- ( suc x e. om -> [_ suc x / x ]_ x = suc x )
88	85, 86, 87	oveq123d 6671	. . . . . . . . . . . . . . 15 |- ( suc x e. om -> ( [_ suc x / x ]_ N [_ suc x / x ]_ +o [_ suc x / x ]_ x ) = ( N +o suc x ) )
89	84, 88	syl5eq 2668	. . . . . . . . . . . . . 14 |- ( suc x e. om -> [_ suc x / x ]_ ( N +o x ) = ( N +o suc x ) )
90	83, 89	fveq12d 6197	. . . . . . . . . . . . 13 |- ( suc x e. om -> ( [_ suc x / x ]_ rec ( F , A ) ` [_ suc x / x ]_ ( N +o x ) ) = ( rec ( F , A ) ` ( N +o suc x ) ) )
91	82, 90	syl5eq 2668	. . . . . . . . . . . 12 |- ( suc x e. om -> [_ suc x / x ]_ ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , A ) ` ( N +o suc x ) ) )
92		csbfv12 6231	. . . . . . . . . . . . 13 |- [_ suc x / x ]_ ( rec ( F , B ) ` ( M +o x ) ) = ( [_ suc x / x ]_ rec ( F , B ) ` [_ suc x / x ]_ ( M +o x ) )
93		csbconstg 3546	. . . . . . . . . . . . . 14 |- ( suc x e. om -> [_ suc x / x ]_ rec ( F , B ) = rec ( F , B ) )
94		csbov123 6687	. . . . . . . . . . . . . . 15 |- [_ suc x / x ]_ ( M +o x ) = ( [_ suc x / x ]_ M [_ suc x / x ]_ +o [_ suc x / x ]_ x )
95		csbconstg 3546	. . . . . . . . . . . . . . . 16 |- ( suc x e. om -> [_ suc x / x ]_ M = M )
96	85, 95, 87	oveq123d 6671	. . . . . . . . . . . . . . 15 |- ( suc x e. om -> ( [_ suc x / x ]_ M [_ suc x / x ]_ +o [_ suc x / x ]_ x ) = ( M +o suc x ) )
97	94, 96	syl5eq 2668	. . . . . . . . . . . . . 14 |- ( suc x e. om -> [_ suc x / x ]_ ( M +o x ) = ( M +o suc x ) )
98	93, 97	fveq12d 6197	. . . . . . . . . . . . 13 |- ( suc x e. om -> ( [_ suc x / x ]_ rec ( F , B ) ` [_ suc x / x ]_ ( M +o x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) )
99	92, 98	syl5eq 2668	. . . . . . . . . . . 12 |- ( suc x e. om -> [_ suc x / x ]_ ( rec ( F , B ) ` ( M +o x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) )
100	91, 99	eqeq12d 2637	. . . . . . . . . . 11 |- ( suc x e. om -> ( [_ suc x / x ]_ ( rec ( F , A ) ` ( N +o x ) ) = [_ suc x / x ]_ ( rec ( F , B ) ` ( M +o x ) ) <-> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) )
101	81, 100	bitrd 268	. . . . . . . . . 10 |- ( suc x e. om -> ( [. suc x / x ]. ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) <-> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) )
102	80, 101	imbi12d 334	. . . . . . . . 9 |- ( suc x e. om -> ( ( [. suc x / x ]. ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> [. suc x / x ]. ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) <-> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) ) )
103	79, 102	bitrd 268	. . . . . . . 8 |- ( suc x e. om -> ( [. suc x / x ]. ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) <-> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) ) )
104	78, 103	imbi12d 334	. . . . . . 7 |- ( suc x e. om -> ( ( [. suc x / x ]. ( N e. On /\ M e. On /\ x e. om ) -> [. suc x / x ]. ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) <-> ( ( N e. On /\ M e. On /\ suc x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) ) ) )
105	71, 104	bitrd 268	. . . . . 6 |- ( suc x e. om -> ( [. suc x / x ]. ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) <-> ( ( N e. On /\ M e. On /\ suc x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o suc x ) ) = ( rec ( F , B ) ` ( M +o suc x ) ) ) ) ) )
106	70, 105	syl5ibr 236	. . . . 5 |- ( suc x e. om -> ( ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) -> [. suc x / x ]. ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) ) )
107	39, 106	sylbi 207	. . . 4 |- ( x e. om -> ( ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) -> [. suc x / x ]. ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) ) )
108	38, 107	findes 7096	. . 3 |- ( x e. om -> ( ( N e. On /\ M e. On /\ x e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o x ) ) = ( rec ( F , B ) ` ( M +o x ) ) ) ) )
109	10, 108	vtoclga 3272	. 2 |- ( X e. om -> ( ( N e. On /\ M e. On /\ X e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o X ) ) = ( rec ( F , B ) ` ( M +o X ) ) ) ) )
110	1, 109	mpcom 38	1 |- ( ( N e. On /\ M e. On /\ X e. om ) -> ( ( rec ( F , A ) ` N ) = ( rec ( F , B ) ` M ) -> ( rec ( F , A ) ` ( N +o X ) ) = ( rec ( F , B ) ` ( M +o X ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   [.wsbc 3435   [_csb 3533   (/)c0 3915   Oncon0 5723   suc csuc 5725   ` cfv 5888  (class class class)co 6650   omcom 7065   reccrdg 7505   +o coa 7557

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-rep 4771  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-fal 1489  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-oadd 7564

This theorem is referenced by:  finxpreclem4  33231