Theorem rexanuz3 39275

Description: Combine two different upper integer properties into one, for a single integer. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
rexanuz3.1	|- F/ j ph
rexanuz3.2	|- Z = ( ZZ>= ` M )
rexanuz3.3	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ch )
rexanuz3.4	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ps )
rexanuz3.5	|- ( k = j -> ( ch <-> th ) )
rexanuz3.6	|- ( k = j -> ( ps <-> ta ) )

Assertion

Ref	Expression
rexanuz3	|- ( ph -> E. j e. Z ( th /\ ta ) )

Distinct variable groups:   j, M   j, Z, k   ch, j   ps, j   ta, k   th, k

Allowed substitution hints:   ph( j, k)   ps( k)   ch( k)   th( j)   ta( j)   M( k)

Proof of Theorem rexanuz3

Step	Hyp	Ref	Expression
1		rexanuz3.3	. . . 4 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ch )
2		rexanuz3.4	. . . 4 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ps )
3	1, 2	jca 554	. . 3 |- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ch /\ E. j e. Z A. k e. ( ZZ>= ` j ) ps ) )
4		rexanuz3.2	. . . 4 |- Z = ( ZZ>= ` M )
5	4	rexanuz2 14089	. . 3 |- ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ch /\ ps ) <-> ( E. j e. Z A. k e. ( ZZ>= ` j ) ch /\ E. j e. Z A. k e. ( ZZ>= ` j ) ps ) )
6	3, 5	sylibr 224	. 2 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( ch /\ ps ) )
7		rexanuz3.1	. . 3 |- F/ j ph
8	4	eleq2i 2693	. . . . . . . . . 10 |- ( j e. Z <-> j e. ( ZZ>= ` M ) )
9	8	biimpi 206	. . . . . . . . 9 |- ( j e. Z -> j e. ( ZZ>= ` M ) )
10		eluzelz 11697	. . . . . . . . 9 |- ( j e. ( ZZ>= ` M ) -> j e. ZZ )
11		uzid 11702	. . . . . . . . 9 |- ( j e. ZZ -> j e. ( ZZ>= ` j ) )
12	9, 10, 11	3syl 18	. . . . . . . 8 |- ( j e. Z -> j e. ( ZZ>= ` j ) )
13	12	adantr 481	. . . . . . 7 |- ( ( j e. Z /\ A. k e. ( ZZ>= ` j ) ( ch /\ ps ) ) -> j e. ( ZZ>= ` j ) )
14		simpr 477	. . . . . . 7 |- ( ( j e. Z /\ A. k e. ( ZZ>= ` j ) ( ch /\ ps ) ) -> A. k e. ( ZZ>= ` j ) ( ch /\ ps ) )
15		rexanuz3.5	. . . . . . . . 9 |- ( k = j -> ( ch <-> th ) )
16		rexanuz3.6	. . . . . . . . 9 |- ( k = j -> ( ps <-> ta ) )
17	15, 16	anbi12d 747	. . . . . . . 8 |- ( k = j -> ( ( ch /\ ps ) <-> ( th /\ ta ) ) )
18	17	rspcva 3307	. . . . . . 7 |- ( ( j e. ( ZZ>= ` j ) /\ A. k e. ( ZZ>= ` j ) ( ch /\ ps ) ) -> ( th /\ ta ) )
19	13, 14, 18	syl2anc 693	. . . . . 6 |- ( ( j e. Z /\ A. k e. ( ZZ>= ` j ) ( ch /\ ps ) ) -> ( th /\ ta ) )
20	19	adantll 750	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ A. k e. ( ZZ>= ` j ) ( ch /\ ps ) ) -> ( th /\ ta ) )
21	20	ex 450	. . . 4 |- ( ( ph /\ j e. Z ) -> ( A. k e. ( ZZ>= ` j ) ( ch /\ ps ) -> ( th /\ ta ) ) )
22	21	ex 450	. . 3 |- ( ph -> ( j e. Z -> ( A. k e. ( ZZ>= ` j ) ( ch /\ ps ) -> ( th /\ ta ) ) ) )
23	7, 22	reximdai 3012	. 2 |- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ch /\ ps ) -> E. j e. Z ( th /\ ta ) ) )
24	6, 23	mpd 15	1 |- ( ph -> E. j e. Z ( th /\ ta ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913   ` cfv 5888   ZZcz 11377   ZZ>=cuz 11687

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  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688

This theorem is referenced by:  smflimlem4  40982