Theorem 4rexfrabdioph 37362

Description: Diophantine set builder for existential quantifier, explicit substitution, four variables. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)

Hypotheses

Ref	Expression
rexfrabdioph.1	|- M = ( N + 1 )
rexfrabdioph.2	|- L = ( M + 1 )
rexfrabdioph.3	|- K = ( L + 1 )
rexfrabdioph.4	|- J = ( K + 1 )

Assertion

Ref	Expression
4rexfrabdioph	|- ( ( N e. NN0 /\ { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph } e. (Dioph ` J ) ) -> { u e. ( NN0 ^m ( 1 ... N ) ) | E. v e. NN0 E. w e. NN0 E. x e. NN0 E. y e. NN0 ph } e. (Dioph ` N ) )

Distinct variable groups:   u, t, v, w, x, y, J   t, K, u, v, w, x, y   t, L, u, v, w, x, y   t, M, u, v, w, x, y   t, N, u, v, w, x, y   ph, t

Allowed substitution hints:   ph( x, y, w, v, u)

Proof of Theorem 4rexfrabdioph

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2sbcrex 37348	. . . . . . . 8 |- ( [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. x e. NN0 E. y e. NN0 ph <-> E. x e. NN0 [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. y e. NN0 ph )
2		2sbcrex 37348	. . . . . . . . 9 |- ( [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. y e. NN0 ph <-> E. y e. NN0 [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph )
3	2	rexbii 3041	. . . . . . . 8 |- ( E. x e. NN0 [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. y e. NN0 ph <-> E. x e. NN0 E. y e. NN0 [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph )
4	1, 3	bitri 264	. . . . . . 7 |- ( [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. x e. NN0 E. y e. NN0 ph <-> E. x e. NN0 E. y e. NN0 [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph )
5	4	sbcbii 3491	. . . . . 6 |- ( [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. x e. NN0 E. y e. NN0 ph <-> [. ( a |` ( 1 ... N ) ) / u ]. E. x e. NN0 E. y e. NN0 [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph )
6		sbc2rex 37351	. . . . . 6 |- ( [. ( a |` ( 1 ... N ) ) / u ]. E. x e. NN0 E. y e. NN0 [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph <-> E. x e. NN0 E. y e. NN0 [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph )
7	5, 6	bitri 264	. . . . 5 |- ( [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. x e. NN0 E. y e. NN0 ph <-> E. x e. NN0 E. y e. NN0 [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph )
8	7	a1i 11	. . . 4 |- ( a e. ( NN0 ^m ( 1 ... L ) ) -> ( [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. x e. NN0 E. y e. NN0 ph <-> E. x e. NN0 E. y e. NN0 [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph ) )
9	8	rabbiia 3185	. . 3 |- { a e. ( NN0 ^m ( 1 ... L ) ) | [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. x e. NN0 E. y e. NN0 ph } = { a e. ( NN0 ^m ( 1 ... L ) ) | E. x e. NN0 E. y e. NN0 [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph }
10		rexfrabdioph.2	. . . . . . 7 |- L = ( M + 1 )
11		rexfrabdioph.1	. . . . . . . . 9 |- M = ( N + 1 )
12		nn0p1nn 11332	. . . . . . . . 9 |- ( N e. NN0 -> ( N + 1 ) e. NN )
13	11, 12	syl5eqel 2705	. . . . . . . 8 |- ( N e. NN0 -> M e. NN )
14	13	peano2nnd 11037	. . . . . . 7 |- ( N e. NN0 -> ( M + 1 ) e. NN )
15	10, 14	syl5eqel 2705	. . . . . 6 |- ( N e. NN0 -> L e. NN )
16	15	nnnn0d 11351	. . . . 5 |- ( N e. NN0 -> L e. NN0 )
17	16	adantr 481	. . . 4 |- ( ( N e. NN0 /\ { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph } e. (Dioph ` J ) ) -> L e. NN0 )
18		sbcrot3 37355	. . . . . . . . . 10 |- ( [. ( a |` ( 1 ... N ) ) / u ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph <-> [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph )
19		sbcrot3 37355	. . . . . . . . . . . . 13 |- ( [. ( t ` J ) / y ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph <-> [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` J ) / y ]. ph )
20	19	sbcbii 3491	. . . . . . . . . . . 12 |- ( [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph <-> [. ( t ` K ) / x ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` J ) / y ]. ph )
21		sbcrot3 37355	. . . . . . . . . . . 12 |- ( [. ( t ` K ) / x ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` J ) / y ]. ph <-> [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph )
22	20, 21	bitri 264	. . . . . . . . . . 11 |- ( [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph <-> [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph )
23	22	sbcbii 3491	. . . . . . . . . 10 |- ( [. ( a |` ( 1 ... N ) ) / u ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph <-> [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph )
24	18, 23	bitr3i 266	. . . . . . . . 9 |- ( [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph <-> [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph )
25	24	sbcbii 3491	. . . . . . . 8 |- ( [. ( t |` ( 1 ... L ) ) / a ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph <-> [. ( t |` ( 1 ... L ) ) / a ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph )
26		reseq1 5390	. . . . . . . . . 10 |- ( a = ( t |` ( 1 ... L ) ) -> ( a |` ( 1 ... N ) ) = ( ( t |` ( 1 ... L ) ) |` ( 1 ... N ) ) )
27	26	sbccomieg 37357	. . . . . . . . 9 |- ( [. ( t |` ( 1 ... L ) ) / a ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( ( t |` ( 1 ... L ) ) |` ( 1 ... N ) ) / u ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph )
28		fzssp1 12384	. . . . . . . . . . . . 13 |- ( 1 ... N ) C_ ( 1 ... ( N + 1 ) )
29	11	oveq2i 6661	. . . . . . . . . . . . 13 |- ( 1 ... M ) = ( 1 ... ( N + 1 ) )
30	28, 29	sseqtr4i 3638	. . . . . . . . . . . 12 |- ( 1 ... N ) C_ ( 1 ... M )
31		fzssp1 12384	. . . . . . . . . . . . 13 |- ( 1 ... M ) C_ ( 1 ... ( M + 1 ) )
32	10	oveq2i 6661	. . . . . . . . . . . . 13 |- ( 1 ... L ) = ( 1 ... ( M + 1 ) )
33	31, 32	sseqtr4i 3638	. . . . . . . . . . . 12 |- ( 1 ... M ) C_ ( 1 ... L )
34	30, 33	sstri 3612	. . . . . . . . . . 11 |- ( 1 ... N ) C_ ( 1 ... L )
35		resabs1 5427	. . . . . . . . . . 11 |- ( ( 1 ... N ) C_ ( 1 ... L ) -> ( ( t |` ( 1 ... L ) ) |` ( 1 ... N ) ) = ( t |` ( 1 ... N ) ) )
36		dfsbcq 3437	. . . . . . . . . . 11 |- ( ( ( t |` ( 1 ... L ) ) |` ( 1 ... N ) ) = ( t |` ( 1 ... N ) ) -> ( [. ( ( t |` ( 1 ... L ) ) |` ( 1 ... N ) ) / u ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t |` ( 1 ... N ) ) / u ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
37	34, 35, 36	mp2b 10	. . . . . . . . . 10 |- ( [. ( ( t |` ( 1 ... L ) ) |` ( 1 ... N ) ) / u ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t |` ( 1 ... N ) ) / u ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph )
38		fveq1 6190	. . . . . . . . . . . . 13 |- ( a = ( t |` ( 1 ... L ) ) -> ( a ` M ) = ( ( t |` ( 1 ... L ) ) ` M ) )
39	38	sbccomieg 37357	. . . . . . . . . . . 12 |- ( [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( ( t |` ( 1 ... L ) ) ` M ) / v ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph )
40		elfz1end 12371	. . . . . . . . . . . . . . . 16 |- ( M e. NN <-> M e. ( 1 ... M ) )
41	13, 40	sylib 208	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> M e. ( 1 ... M ) )
42	33, 41	sseldi 3601	. . . . . . . . . . . . . 14 |- ( N e. NN0 -> M e. ( 1 ... L ) )
43		fvres 6207	. . . . . . . . . . . . . 14 |- ( M e. ( 1 ... L ) -> ( ( t |` ( 1 ... L ) ) ` M ) = ( t ` M ) )
44		dfsbcq 3437	. . . . . . . . . . . . . 14 |- ( ( ( t |` ( 1 ... L ) ) ` M ) = ( t ` M ) -> ( [. ( ( t |` ( 1 ... L ) ) ` M ) / v ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t ` M ) / v ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
45	42, 43, 44	3syl 18	. . . . . . . . . . . . 13 |- ( N e. NN0 -> ( [. ( ( t |` ( 1 ... L ) ) ` M ) / v ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t ` M ) / v ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
46		vex 3203	. . . . . . . . . . . . . . . . 17 |- t e. _V
47	46	resex 5443	. . . . . . . . . . . . . . . 16 |- ( t |` ( 1 ... L ) ) e. _V
48		fveq1 6190	. . . . . . . . . . . . . . . . 17 |- ( a = ( t |` ( 1 ... L ) ) -> ( a ` L ) = ( ( t |` ( 1 ... L ) ) ` L ) )
49	48	sbcco3g 3999	. . . . . . . . . . . . . . . 16 |- ( ( t |` ( 1 ... L ) ) e. _V -> ( [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( ( t |` ( 1 ... L ) ) ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
50	47, 49	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( ( t |` ( 1 ... L ) ) ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph )
51		elfz1end 12371	. . . . . . . . . . . . . . . . 17 |- ( L e. NN <-> L e. ( 1 ... L ) )
52	15, 51	sylib 208	. . . . . . . . . . . . . . . 16 |- ( N e. NN0 -> L e. ( 1 ... L ) )
53		fvres 6207	. . . . . . . . . . . . . . . 16 |- ( L e. ( 1 ... L ) -> ( ( t |` ( 1 ... L ) ) ` L ) = ( t ` L ) )
54		dfsbcq 3437	. . . . . . . . . . . . . . . 16 |- ( ( ( t |` ( 1 ... L ) ) ` L ) = ( t ` L ) -> ( [. ( ( t |` ( 1 ... L ) ) ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
55	52, 53, 54	3syl 18	. . . . . . . . . . . . . . 15 |- ( N e. NN0 -> ( [. ( ( t |` ( 1 ... L ) ) ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
56	50, 55	syl5bb 272	. . . . . . . . . . . . . 14 |- ( N e. NN0 -> ( [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
57	56	sbcbidv 3490	. . . . . . . . . . . . 13 |- ( N e. NN0 -> ( [. ( t ` M ) / v ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
58	45, 57	bitrd 268	. . . . . . . . . . . 12 |- ( N e. NN0 -> ( [. ( ( t |` ( 1 ... L ) ) ` M ) / v ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
59	39, 58	syl5bb 272	. . . . . . . . . . 11 |- ( N e. NN0 -> ( [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
60	59	sbcbidv 3490	. . . . . . . . . 10 |- ( N e. NN0 -> ( [. ( t |` ( 1 ... N ) ) / u ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
61	37, 60	syl5bb 272	. . . . . . . . 9 |- ( N e. NN0 -> ( [. ( ( t |` ( 1 ... L ) ) |` ( 1 ... N ) ) / u ]. [. ( t |` ( 1 ... L ) ) / a ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
62	27, 61	syl5bb 272	. . . . . . . 8 |- ( N e. NN0 -> ( [. ( t |` ( 1 ... L ) ) / a ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph <-> [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
63	25, 62	syl5bb 272	. . . . . . 7 |- ( N e. NN0 -> ( [. ( t |` ( 1 ... L ) ) / a ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph <-> [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph ) )
64	63	rabbidv 3189	. . . . . 6 |- ( N e. NN0 -> { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... L ) ) / a ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph } = { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph } )
65	64	eleq1d 2686	. . . . 5 |- ( N e. NN0 -> ( { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... L ) ) / a ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph } e. (Dioph ` J ) <-> { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph } e. (Dioph ` J ) ) )
66	65	biimpar 502	. . . 4 |- ( ( N e. NN0 /\ { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph } e. (Dioph ` J ) ) -> { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... L ) ) / a ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph } e. (Dioph ` J ) )
67		rexfrabdioph.3	. . . . 5 |- K = ( L + 1 )
68		rexfrabdioph.4	. . . . 5 |- J = ( K + 1 )
69	67, 68	2rexfrabdioph 37360	. . . 4 |- ( ( L e. NN0 /\ { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... L ) ) / a ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph } e. (Dioph ` J ) ) -> { a e. ( NN0 ^m ( 1 ... L ) ) | E. x e. NN0 E. y e. NN0 [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph } e. (Dioph ` L ) )
70	17, 66, 69	syl2anc 693	. . 3 |- ( ( N e. NN0 /\ { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph } e. (Dioph ` J ) ) -> { a e. ( NN0 ^m ( 1 ... L ) ) | E. x e. NN0 E. y e. NN0 [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. ph } e. (Dioph ` L ) )
71	9, 70	syl5eqel 2705	. 2 |- ( ( N e. NN0 /\ { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph } e. (Dioph ` J ) ) -> { a e. ( NN0 ^m ( 1 ... L ) ) | [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. x e. NN0 E. y e. NN0 ph } e. (Dioph ` L ) )
72	11, 10	2rexfrabdioph 37360	. 2 |- ( ( N e. NN0 /\ { a e. ( NN0 ^m ( 1 ... L ) ) | [. ( a |` ( 1 ... N ) ) / u ]. [. ( a ` M ) / v ]. [. ( a ` L ) / w ]. E. x e. NN0 E. y e. NN0 ph } e. (Dioph ` L ) ) -> { u e. ( NN0 ^m ( 1 ... N ) ) | E. v e. NN0 E. w e. NN0 E. x e. NN0 E. y e. NN0 ph } e. (Dioph ` N ) )
73	71, 72	syldan 487	1 |- ( ( N e. NN0 /\ { t e. ( NN0 ^m ( 1 ... J ) ) | [. ( t |` ( 1 ... N ) ) / u ]. [. ( t ` M ) / v ]. [. ( t ` L ) / w ]. [. ( t ` K ) / x ]. [. ( t ` J ) / y ]. ph } e. (Dioph ` J ) ) -> { u e. ( NN0 ^m ( 1 ... N ) ) | E. v e. NN0 E. w e. NN0 E. x e. NN0 E. y e. NN0 ph } e. (Dioph ` N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   [.wsbc 3435   C_ wss 3574   |` cres 5116   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   1c1 9937   + caddc 9939   NNcn 11020   NN0cn0 11292   ...cfz 12326  Diophcdioph 37318

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  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-int 4476  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-mzpcl 37286  df-mzp 37287  df-dioph 37319

This theorem is referenced by:  6rexfrabdioph  37363