Theorem frec2uzrdg 8876

Description: A helper lemma for the value of a recursive definition generator on upper integers (typically either NN or NN0) with characteristic function F(x , y) and initial value A. This lemma shows that evaluating R at an element of om gives an ordered pair whose first element is the index (translated from om to ( ZZ>= ` C)). See comment in frec2uz0d 8866 which describes G and the index translation. (Contributed by Jim Kingdon, 24-May-2020.)

Hypotheses

Ref	Expression
frec2uz.1	|- (ph -> C e. ZZ)
frec2uz.2	|- G = frec((x e. ZZ |-> (x + 1)) , C)
uzrdg.s	|- (ph -> S e. V)
uzrdg.a	|- (ph -> A e. S)
uzrdg.f	|- ((ph /\ (x e. ( ZZ>= ` C) /\ y e. S)) -> (x Fy) e. S)
uzrdg.2	|- R = frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.)
uzrdg.b	|- (ph -> B e. om)

Assertion

Ref	Expression
frec2uzrdg	|- (ph -> ( R ` B) = <.( G ` B) , ( 2nd ` ( R ` B)) >.)

Distinct variable groups:   y,A   x, C,y   y, G   x, F,y   x, S,y   ph,x,y

Allowed substitution hints:   A(x)   B(x,y)   R(x,y)   G(x)   V(x,y)

Proof of Theorem frec2uzrdg

Dummy variables w z v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzrdg.b	. 2 |- (ph -> B e. om)
2		fveq2 5121	. . . . 5 |- (z = B -> ( R ` z) = ( R ` B))
3		fveq2 5121	. . . . . 6 |- (z = B -> ( G ` z) = ( G ` B))
4	2	fveq2d 5125	. . . . . 6 |- (z = B -> ( 2nd ` ( R ` z)) = ( 2nd ` ( R ` B)))
5	3, 4	opeq12d 3548	. . . . 5 |- (z = B -> <.( G ` z) , ( 2nd ` ( R ` z)) >. = <.( G ` B) , ( 2nd ` ( R ` B)) >.)
6	2, 5	eqeq12d 2051	. . . 4 |- (z = B -> (( R ` z) = <.( G ` z) , ( 2nd ` ( R ` z)) >. <-> ( R ` B) = <.( G ` B) , ( 2nd ` ( R ` B)) >.))
7	6	imbi2d 219	. . 3 |- (z = B -> ((ph -> ( R ` z) = <.( G ` z) , ( 2nd ` ( R ` z)) >.) <-> (ph -> ( R ` B) = <.( G ` B) , ( 2nd ` ( R ` B)) >.)))
8		fveq2 5121	. . . . 5 |- (z = (/) -> ( R ` z) = ( R ` (/)))
9		fveq2 5121	. . . . . 6 |- (z = (/) -> ( G ` z) = ( G ` (/)))
10	8	fveq2d 5125	. . . . . 6 |- (z = (/) -> ( 2nd ` ( R ` z)) = ( 2nd ` ( R ` (/))))
11	9, 10	opeq12d 3548	. . . . 5 |- (z = (/) -> <.( G ` z) , ( 2nd ` ( R ` z)) >. = <.( G ` (/)) , ( 2nd ` ( R ` (/))) >.)
12	8, 11	eqeq12d 2051	. . . 4 |- (z = (/) -> (( R ` z) = <.( G ` z) , ( 2nd ` ( R ` z)) >. <-> ( R ` (/)) = <.( G ` (/)) , ( 2nd ` ( R ` (/))) >.))
13		fveq2 5121	. . . . 5 |- (z = v -> ( R ` z) = ( R ` v))
14		fveq2 5121	. . . . . 6 |- (z = v -> ( G ` z) = ( G ` v))
15	13	fveq2d 5125	. . . . . 6 |- (z = v -> ( 2nd ` ( R ` z)) = ( 2nd ` ( R ` v)))
16	14, 15	opeq12d 3548	. . . . 5 |- (z = v -> <.( G ` z) , ( 2nd ` ( R ` z)) >. = <.( G ` v) , ( 2nd ` ( R ` v)) >.)
17	13, 16	eqeq12d 2051	. . . 4 |- (z = v -> (( R ` z) = <.( G ` z) , ( 2nd ` ( R ` z)) >. <-> ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.))
18		fveq2 5121	. . . . 5 |- (z = suc v -> ( R ` z) = ( R ` suc v))
19		fveq2 5121	. . . . . 6 |- (z = suc v -> ( G ` z) = ( G ` suc v))
20	18	fveq2d 5125	. . . . . 6 |- (z = suc v -> ( 2nd ` ( R ` z)) = ( 2nd ` ( R ` suc v)))
21	19, 20	opeq12d 3548	. . . . 5 |- (z = suc v -> <.( G ` z) , ( 2nd ` ( R ` z)) >. = <.( G ` suc v) , ( 2nd ` ( R ` suc v)) >.)
22	18, 21	eqeq12d 2051	. . . 4 |- (z = suc v -> (( R ` z) = <.( G ` z) , ( 2nd ` ( R ` z)) >. <-> ( R ` suc v) = <.( G ` suc v) , ( 2nd ` ( R ` suc v)) >.))
23		uzrdg.2	. . . . . . 7 |- R = frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.)
24	23	fveq1i 5122	. . . . . 6 |- ( R ` (/)) = (frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.) ` (/))
25		frec2uz.1	. . . . . . . 8 |- (ph -> C e. ZZ)
26		uzrdg.a	. . . . . . . 8 |- (ph -> A e. S)
27		opexg 3955	. . . . . . . 8 |- (( C e. ZZ /\ A e. S) -> <. C , A >. e. _V)
28	25, 26, 27	syl2anc 391	. . . . . . 7 |- (ph -> <. C , A >. e. _V)
29		frec0g 5922	. . . . . . 7 |- ( <. C , A >. e. _V -> (frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.) ` (/)) = <. C , A >.)
30	28, 29	syl 14	. . . . . 6 |- (ph -> (frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.) ` (/)) = <. C , A >.)
31	24, 30	syl5eq 2081	. . . . 5 |- (ph -> ( R ` (/)) = <. C , A >.)
32		frec2uz.2	. . . . . . 7 |- G = frec((x e. ZZ |-> (x + 1)) , C)
33	25, 32	frec2uz0d 8866	. . . . . 6 |- (ph -> ( G ` (/)) = C)
34	31	fveq2d 5125	. . . . . . 7 |- (ph -> ( 2nd ` ( R ` (/))) = ( 2nd ` <. C , A >.))
35		uzid 8263	. . . . . . . . 9 |- ( C e. ZZ -> C e. ( ZZ>= ` C))
36	25, 35	syl 14	. . . . . . . 8 |- (ph -> C e. ( ZZ>= ` C))
37		op2ndg 5720	. . . . . . . 8 |- (( C e. ( ZZ>= ` C) /\ A e. S) -> ( 2nd ` <. C , A >.) = A)
38	36, 26, 37	syl2anc 391	. . . . . . 7 |- (ph -> ( 2nd ` <. C , A >.) = A)
39	34, 38	eqtrd 2069	. . . . . 6 |- (ph -> ( 2nd ` ( R ` (/))) = A)
40	33, 39	opeq12d 3548	. . . . 5 |- (ph -> <.( G ` (/)) , ( 2nd ` ( R ` (/))) >. = <. C , A >.)
41	31, 40	eqtr4d 2072	. . . 4 |- (ph -> ( R ` (/)) = <.( G ` (/)) , ( 2nd ` ( R ` (/))) >.)
42		zex 8030	. . . . . . . . . . . . . . . 16 |- ZZ e. _V
43		uzssz 8268	. . . . . . . . . . . . . . . 16 |- ( ZZ>= ` C) C_ ZZ
44	42, 43	ssexi 3886	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` C) e. _V
45	44	a1i 9	. . . . . . . . . . . . . 14 |- ((ph /\ v e. om) -> ( ZZ>= ` C) e. _V)
46		uzrdg.s	. . . . . . . . . . . . . . 15 |- (ph -> S e. V)
47	46	adantr 261	. . . . . . . . . . . . . 14 |- ((ph /\ v e. om) -> S e. V)
48		mpt2exga 5777	. . . . . . . . . . . . . 14 |- ((( ZZ>= ` C) e. _V /\ S e. V) -> (x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) e. _V)
49	45, 47, 48	syl2anc 391	. . . . . . . . . . . . 13 |- ((ph /\ v e. om) -> (x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) e. _V)
50		vex 2554	. . . . . . . . . . . . . 14 |- z e. _V
51	50	a1i 9	. . . . . . . . . . . . 13 |- ((ph /\ v e. om) -> z e. _V)
52		fvexg 5137	. . . . . . . . . . . . 13 |- (((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) e. _V /\ z e. _V) -> ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` z) e. _V)
53	49, 51, 52	syl2anc 391	. . . . . . . . . . . 12 |- ((ph /\ v e. om) -> ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` z) e. _V)
54	53	alrimiv 1751	. . . . . . . . . . 11 |- ((ph /\ v e. om) -> A.z((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` z) e. _V)
55	28	adantr 261	. . . . . . . . . . 11 |- ((ph /\ v e. om) -> <. C , A >. e. _V)
56		simpr 103	. . . . . . . . . . 11 |- ((ph /\ v e. om) -> v e. om)
57		frecsuc 5930	. . . . . . . . . . 11 |- ((A.z((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` z) e. _V /\ <. C , A >. e. _V /\ v e. om) -> (frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.) ` suc v) = ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` (frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.) ` v)))
58	54, 55, 56, 57	syl3anc 1134	. . . . . . . . . 10 |- ((ph /\ v e. om) -> (frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.) ` suc v) = ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` (frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.) ` v)))
59	23	fveq1i 5122	. . . . . . . . . 10 |- ( R ` suc v) = (frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.) ` suc v)
60	23	fveq1i 5122	. . . . . . . . . . 11 |- ( R ` v) = (frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.) ` v)
61	60	fveq2i 5124	. . . . . . . . . 10 |- ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` ( R ` v)) = ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` (frec((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) , <. C , A >.) ` v))
62	58, 59, 61	3eqtr4g 2094	. . . . . . . . 9 |- ((ph /\ v e. om) -> ( R ` suc v) = ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` ( R ` v)))
63	62	adantr 261	. . . . . . . 8 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> ( R ` suc v) = ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` ( R ` v)))
64		fveq2 5121	. . . . . . . . 9 |- (( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >. -> ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` ( R ` v)) = ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` <.( G ` v) , ( 2nd ` ( R ` v)) >.))
65		df-ov 5458	. . . . . . . . . 10 |- (( G ` v)(x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.)( 2nd ` ( R ` v))) = ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` <.( G ` v) , ( 2nd ` ( R ` v)) >.)
66	25	adantr 261	. . . . . . . . . . . 12 |- ((ph /\ v e. om) -> C e. ZZ)
67	66, 32, 56	frec2uzuzd 8869	. . . . . . . . . . 11 |- ((ph /\ v e. om) -> ( G ` v) e. ( ZZ>= ` C))
68		uzrdg.f	. . . . . . . . . . . . 13 |- ((ph /\ (x e. ( ZZ>= ` C) /\ y e. S)) -> (x Fy) e. S)
69	25, 32, 46, 26, 68, 23	frecuzrdgrrn 8875	. . . . . . . . . . . 12 |- ((ph /\ v e. om) -> ( R ` v) e. (( ZZ>= ` C) X. S))
70		xp2nd 5735	. . . . . . . . . . . 12 |- (( R ` v) e. (( ZZ>= ` C) X. S) -> ( 2nd ` ( R ` v)) e. S)
71	69, 70	syl 14	. . . . . . . . . . 11 |- ((ph /\ v e. om) -> ( 2nd ` ( R ` v)) e. S)
72		peano2uz 8302	. . . . . . . . . . . . 13 |- (( G ` v) e. ( ZZ>= ` C) -> (( G ` v) + 1) e. ( ZZ>= ` C))
73	67, 72	syl 14	. . . . . . . . . . . 12 |- ((ph /\ v e. om) -> (( G ` v) + 1) e. ( ZZ>= ` C))
74	68	caovclg 5595	. . . . . . . . . . . . . 14 |- ((ph /\ (z e. ( ZZ>= ` C) /\ w e. S)) -> (z Fw) e. S)
75	74	adantlr 446	. . . . . . . . . . . . 13 |- (((ph /\ v e. om) /\ (z e. ( ZZ>= ` C) /\ w e. S)) -> (z Fw) e. S)
76	75, 67, 71	caovcld 5596	. . . . . . . . . . . 12 |- ((ph /\ v e. om) -> (( G ` v) F( 2nd ` ( R ` v))) e. S)
77		opelxp 4317	. . . . . . . . . . . 12 |- ( <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >. e. (( ZZ>= ` C) X. S) <-> ((( G ` v) + 1) e. ( ZZ>= ` C) /\ (( G ` v) F( 2nd ` ( R ` v))) e. S))
78	73, 76, 77	sylanbrc 394	. . . . . . . . . . 11 |- ((ph /\ v e. om) -> <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >. e. (( ZZ>= ` C) X. S))
79		oveq1 5462	. . . . . . . . . . . . 13 |- (w = ( G ` v) -> (w + 1) = (( G ` v) + 1))
80		oveq1 5462	. . . . . . . . . . . . 13 |- (w = ( G ` v) -> (w Fz) = (( G ` v) Fz))
81	79, 80	opeq12d 3548	. . . . . . . . . . . 12 |- (w = ( G ` v) -> <.(w + 1) , (w Fz) >. = <.(( G ` v) + 1) , (( G ` v) Fz) >.)
82		oveq2 5463	. . . . . . . . . . . . 13 |- (z = ( 2nd ` ( R ` v)) -> (( G ` v) Fz) = (( G ` v) F( 2nd ` ( R ` v))))
83	82	opeq2d 3547	. . . . . . . . . . . 12 |- (z = ( 2nd ` ( R ` v)) -> <.(( G ` v) + 1) , (( G ` v) Fz) >. = <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >.)
84		oveq1 5462	. . . . . . . . . . . . . 14 |- (x = w -> (x + 1) = (w + 1))
85		oveq1 5462	. . . . . . . . . . . . . 14 |- (x = w -> (x Fy) = (w Fy))
86	84, 85	opeq12d 3548	. . . . . . . . . . . . 13 |- (x = w -> <.(x + 1) , (x Fy) >. = <.(w + 1) , (w Fy) >.)
87		oveq2 5463	. . . . . . . . . . . . . 14 |- (y = z -> (w Fy) = (w Fz))
88	87	opeq2d 3547	. . . . . . . . . . . . 13 |- (y = z -> <.(w + 1) , (w Fy) >. = <.(w + 1) , (w Fz) >.)
89	86, 88	cbvmpt2v 5526	. . . . . . . . . . . 12 |- (x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) = (w e. ( ZZ>= ` C) , z e. S |-> <.(w + 1) , (w Fz) >.)
90	81, 83, 89	ovmpt2g 5577	. . . . . . . . . . 11 |- ((( G ` v) e. ( ZZ>= ` C) /\ ( 2nd ` ( R ` v)) e. S /\ <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >. e. (( ZZ>= ` C) X. S)) -> (( G ` v)(x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.)( 2nd ` ( R ` v))) = <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >.)
91	67, 71, 78, 90	syl3anc 1134	. . . . . . . . . 10 |- ((ph /\ v e. om) -> (( G ` v)(x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.)( 2nd ` ( R ` v))) = <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >.)
92	65, 91	syl5eqr 2083	. . . . . . . . 9 |- ((ph /\ v e. om) -> ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` <.( G ` v) , ( 2nd ` ( R ` v)) >.) = <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >.)
93	64, 92	sylan9eqr 2091	. . . . . . . 8 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> ((x e. ( ZZ>= ` C) , y e. S |-> <.(x + 1) , (x Fy) >.) ` ( R ` v)) = <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >.)
94	63, 93	eqtrd 2069	. . . . . . 7 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> ( R ` suc v) = <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >.)
95	66, 32, 56	frec2uzsucd 8868	. . . . . . . . 9 |- ((ph /\ v e. om) -> ( G ` suc v) = (( G ` v) + 1))
96	95	adantr 261	. . . . . . . 8 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> ( G ` suc v) = (( G ` v) + 1))
97	94	fveq2d 5125	. . . . . . . . 9 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> ( 2nd ` ( R ` suc v)) = ( 2nd ` <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >.))
98	66, 32, 56	frec2uzzd 8867	. . . . . . . . . . . 12 |- ((ph /\ v e. om) -> ( G ` v) e. ZZ)
99	98	peano2zd 8139	. . . . . . . . . . 11 |- ((ph /\ v e. om) -> (( G ` v) + 1) e. ZZ)
100	99	adantr 261	. . . . . . . . . 10 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> (( G ` v) + 1) e. ZZ)
101	76	adantr 261	. . . . . . . . . 10 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> (( G ` v) F( 2nd ` ( R ` v))) e. S)
102		op2ndg 5720	. . . . . . . . . 10 |- (((( G ` v) + 1) e. ZZ /\ (( G ` v) F( 2nd ` ( R ` v))) e. S) -> ( 2nd ` <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >.) = (( G ` v) F( 2nd ` ( R ` v))))
103	100, 101, 102	syl2anc 391	. . . . . . . . 9 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> ( 2nd ` <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >.) = (( G ` v) F( 2nd ` ( R ` v))))
104	97, 103	eqtrd 2069	. . . . . . . 8 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> ( 2nd ` ( R ` suc v)) = (( G ` v) F( 2nd ` ( R ` v))))
105	96, 104	opeq12d 3548	. . . . . . 7 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> <.( G ` suc v) , ( 2nd ` ( R ` suc v)) >. = <.(( G ` v) + 1) , (( G ` v) F( 2nd ` ( R ` v))) >.)
106	94, 105	eqtr4d 2072	. . . . . 6 |- (((ph /\ v e. om) /\ ( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >.) -> ( R ` suc v) = <.( G ` suc v) , ( 2nd ` ( R ` suc v)) >.)
107	106	ex 108	. . . . 5 |- ((ph /\ v e. om) -> (( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >. -> ( R ` suc v) = <.( G ` suc v) , ( 2nd ` ( R ` suc v)) >.))
108	107	expcom 109	. . . 4 |- (v e. om -> (ph -> (( R ` v) = <.( G ` v) , ( 2nd ` ( R ` v)) >. -> ( R ` suc v) = <.( G ` suc v) , ( 2nd ` ( R ` suc v)) >.)))
109	12, 17, 22, 41, 108	finds2 4267	. . 3 |- (z e. om -> (ph -> ( R ` z) = <.( G ` z) , ( 2nd ` ( R ` z)) >.))
110	7, 109	vtoclga 2613	. 2 |- (B e. om -> (ph -> ( R ` B) = <.( G ` B) , ( 2nd ` ( R ` B)) >.))
111	1, 110	mpcom 32	1 |- (ph -> ( R ` B) = <.( G ` B) , ( 2nd ` ( R ` B)) >.)

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240   = wceq 1242   e. wcel 1390   _Vcvv 2551   (/)c0 3218   <.cop 3370   |-> cmpt 3809   suc csuc 4068   omcom 4256   X. cxp 4286   ` cfv 4845  (class class class)co 5455   |-> cmpt2 5457   2ndc2nd 5708  freccfrec 5917   1c1 6712   + caddc 6714   ZZcz 8021   ZZ>=cuz 8249

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-distr 6787  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797  ax-pre-ltadd 6799

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-frec 5918  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-sub 6981  df-neg 6982  df-inn 7696  df-n0 7958  df-z 8022  df-uz 8250

This theorem is referenced by:  frecuzrdglem  8878  frecuzrdgfn  8879  frecuzrdgsuc  8882