Theorem ntrivcvgmullem 14633

Description: Lemma for ntrivcvgmul 14634. (Contributed by Scott Fenton, 19-Dec-2017.)

Hypotheses

Ref	Expression
ntrivcvgmullem.1	|- Z = ( ZZ>= ` M )
ntrivcvgmullem.2	|- ( ph -> N e. Z )
ntrivcvgmullem.3	|- ( ph -> P e. Z )
ntrivcvgmullem.4	|- ( ph -> X =/= 0 )
ntrivcvgmullem.5	|- ( ph -> Y =/= 0 )
ntrivcvgmullem.6	|- ( ph -> seq N ( x. , F ) ~~> X )
ntrivcvgmullem.7	|- ( ph -> seq P ( x. , G ) ~~> Y )
ntrivcvgmullem.8	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
ntrivcvgmullem.9	|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
ntrivcvgmullem.a	|- ( ph -> N <_ P )
ntrivcvgmullem.b	|- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )

Assertion

Ref	Expression
ntrivcvgmullem	|- ( ph -> E. q e. Z E. w ( w =/= 0 /\ seq q ( x. , H ) ~~> w ) )

Distinct variable groups:   w, F   H, q, w   P, q, w   w, Y   Z, q   k, F   k, G   k, H   ph, k   P, k   k, Z   k, N

Allowed substitution hints:   ph( w, q)   F( q)   G( w, q)   M( w, k, q)   N( w, q)   X( w, k, q)   Y( k, q)   Z( w)

Proof of Theorem ntrivcvgmullem

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ntrivcvgmullem.3	. 2 |- ( ph -> P e. Z )
2		eqid 2622	. . . . . . 7 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
3		ntrivcvgmullem.a	. . . . . . . 8 |- ( ph -> N <_ P )
4		ntrivcvgmullem.1	. . . . . . . . . . 11 |- Z = ( ZZ>= ` M )
5		uzssz 11707	. . . . . . . . . . 11 |- ( ZZ>= ` M ) C_ ZZ
6	4, 5	eqsstri 3635	. . . . . . . . . 10 |- Z C_ ZZ
7		ntrivcvgmullem.2	. . . . . . . . . 10 |- ( ph -> N e. Z )
8	6, 7	sseldi 3601	. . . . . . . . 9 |- ( ph -> N e. ZZ )
9	6, 1	sseldi 3601	. . . . . . . . 9 |- ( ph -> P e. ZZ )
10		eluz 11701	. . . . . . . . 9 |- ( ( N e. ZZ /\ P e. ZZ ) -> ( P e. ( ZZ>= ` N ) <-> N <_ P ) )
11	8, 9, 10	syl2anc 693	. . . . . . . 8 |- ( ph -> ( P e. ( ZZ>= ` N ) <-> N <_ P ) )
12	3, 11	mpbird 247	. . . . . . 7 |- ( ph -> P e. ( ZZ>= ` N ) )
13		ntrivcvgmullem.6	. . . . . . 7 |- ( ph -> seq N ( x. , F ) ~~> X )
14		ntrivcvgmullem.4	. . . . . . 7 |- ( ph -> X =/= 0 )
15	4	uztrn2 11705	. . . . . . . . 9 |- ( ( N e. Z /\ k e. ( ZZ>= ` N ) ) -> k e. Z )
16	7, 15	sylan 488	. . . . . . . 8 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. Z )
17		ntrivcvgmullem.8	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
18	16, 17	syldan 487	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( F ` k ) e. CC )
19	2, 12, 13, 14, 18	ntrivcvgtail 14632	. . . . . 6 |- ( ph -> ( ( ~~> ` seq P ( x. , F ) ) =/= 0 /\ seq P ( x. , F ) ~~> ( ~~> ` seq P ( x. , F ) ) ) )
20	19	simprd 479	. . . . 5 |- ( ph -> seq P ( x. , F ) ~~> ( ~~> ` seq P ( x. , F ) ) )
21		climcl 14230	. . . . 5 |- ( seq P ( x. , F ) ~~> ( ~~> ` seq P ( x. , F ) ) -> ( ~~> ` seq P ( x. , F ) ) e. CC )
22	20, 21	syl 17	. . . 4 |- ( ph -> ( ~~> ` seq P ( x. , F ) ) e. CC )
23		ntrivcvgmullem.7	. . . . 5 |- ( ph -> seq P ( x. , G ) ~~> Y )
24		climcl 14230	. . . . 5 |- ( seq P ( x. , G ) ~~> Y -> Y e. CC )
25	23, 24	syl 17	. . . 4 |- ( ph -> Y e. CC )
26	19	simpld 475	. . . 4 |- ( ph -> ( ~~> ` seq P ( x. , F ) ) =/= 0 )
27		ntrivcvgmullem.5	. . . 4 |- ( ph -> Y =/= 0 )
28	22, 25, 26, 27	mulne0d 10679	. . 3 |- ( ph -> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) =/= 0 )
29		eqid 2622	. . . 4 |- ( ZZ>= ` P ) = ( ZZ>= ` P )
30		seqex 12803	. . . . 5 |- seq P ( x. , H ) e. _V
31	30	a1i 11	. . . 4 |- ( ph -> seq P ( x. , H ) e. _V )
32	4	uztrn2 11705	. . . . . . . 8 |- ( ( P e. Z /\ k e. ( ZZ>= ` P ) ) -> k e. Z )
33	1, 32	sylan 488	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` P ) ) -> k e. Z )
34	33, 17	syldan 487	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` P ) ) -> ( F ` k ) e. CC )
35	29, 9, 34	prodf 14619	. . . . 5 |- ( ph -> seq P ( x. , F ) : ( ZZ>= ` P ) --> CC )
36	35	ffvelrnda 6359	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` P ) ) -> ( seq P ( x. , F ) ` j ) e. CC )
37		ntrivcvgmullem.9	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
38	33, 37	syldan 487	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` P ) ) -> ( G ` k ) e. CC )
39	29, 9, 38	prodf 14619	. . . . 5 |- ( ph -> seq P ( x. , G ) : ( ZZ>= ` P ) --> CC )
40	39	ffvelrnda 6359	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` P ) ) -> ( seq P ( x. , G ) ` j ) e. CC )
41		simpr 477	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` P ) ) -> j e. ( ZZ>= ` P ) )
42		simpll 790	. . . . . 6 |- ( ( ( ph /\ j e. ( ZZ>= ` P ) ) /\ k e. ( P ... j ) ) -> ph )
43	1	adantr 481	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` P ) ) -> P e. Z )
44		elfzuz 12338	. . . . . . 7 |- ( k e. ( P ... j ) -> k e. ( ZZ>= ` P ) )
45	43, 44, 32	syl2an 494	. . . . . 6 |- ( ( ( ph /\ j e. ( ZZ>= ` P ) ) /\ k e. ( P ... j ) ) -> k e. Z )
46	42, 45, 17	syl2anc 693	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` P ) ) /\ k e. ( P ... j ) ) -> ( F ` k ) e. CC )
47	42, 45, 37	syl2anc 693	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` P ) ) /\ k e. ( P ... j ) ) -> ( G ` k ) e. CC )
48		ntrivcvgmullem.b	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
49	42, 45, 48	syl2anc 693	. . . . 5 |- ( ( ( ph /\ j e. ( ZZ>= ` P ) ) /\ k e. ( P ... j ) ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
50	41, 46, 47, 49	prodfmul 14622	. . . 4 |- ( ( ph /\ j e. ( ZZ>= ` P ) ) -> ( seq P ( x. , H ) ` j ) = ( ( seq P ( x. , F ) ` j ) x. ( seq P ( x. , G ) ` j ) ) )
51	29, 9, 20, 31, 23, 36, 40, 50	climmul 14363	. . 3 |- ( ph -> seq P ( x. , H ) ~~> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) )
52		ovex 6678	. . . 4 |- ( ( ~~> ` seq P ( x. , F ) ) x. Y ) e. _V
53		neeq1 2856	. . . . 5 |- ( w = ( ( ~~> ` seq P ( x. , F ) ) x. Y ) -> ( w =/= 0 <-> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) =/= 0 ) )
54		breq2 4657	. . . . 5 |- ( w = ( ( ~~> ` seq P ( x. , F ) ) x. Y ) -> ( seq P ( x. , H ) ~~> w <-> seq P ( x. , H ) ~~> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) ) )
55	53, 54	anbi12d 747	. . . 4 |- ( w = ( ( ~~> ` seq P ( x. , F ) ) x. Y ) -> ( ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) <-> ( ( ( ~~> ` seq P ( x. , F ) ) x. Y ) =/= 0 /\ seq P ( x. , H ) ~~> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) ) ) )
56	52, 55	spcev 3300	. . 3 |- ( ( ( ( ~~> ` seq P ( x. , F ) ) x. Y ) =/= 0 /\ seq P ( x. , H ) ~~> ( ( ~~> ` seq P ( x. , F ) ) x. Y ) ) -> E. w ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) )
57	28, 51, 56	syl2anc 693	. 2 |- ( ph -> E. w ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) )
58		seqeq1 12804	. . . . . 6 |- ( q = P -> seq q ( x. , H ) = seq P ( x. , H ) )
59	58	breq1d 4663	. . . . 5 |- ( q = P -> ( seq q ( x. , H ) ~~> w <-> seq P ( x. , H ) ~~> w ) )
60	59	anbi2d 740	. . . 4 |- ( q = P -> ( ( w =/= 0 /\ seq q ( x. , H ) ~~> w ) <-> ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) ) )
61	60	exbidv 1850	. . 3 |- ( q = P -> ( E. w ( w =/= 0 /\ seq q ( x. , H ) ~~> w ) <-> E. w ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) ) )
62	61	rspcev 3309	. 2 |- ( ( P e. Z /\ E. w ( w =/= 0 /\ seq P ( x. , H ) ~~> w ) ) -> E. q e. Z E. w ( w =/= 0 /\ seq q ( x. , H ) ~~> w ) )
63	1, 57, 62	syl2anc 693	1 |- ( ph -> E. q e. Z E. w ( w =/= 0 /\ seq q ( x. , H ) ~~> w ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215

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  ax-pre-sup 10014

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219

This theorem is referenced by:  ntrivcvgmul  14634