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Theorem ntrivcvgmul 14634
Description: The product of two non-trivially converging products converges non-trivially. (Contributed by Scott Fenton, 18-Dec-2017.)
Hypotheses
Ref Expression
ntrivcvgmul.1 |- Z = ( ZZ>= ` M )
ntrivcvgmul.3 |- ( ph -> E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) )
ntrivcvgmul.4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
ntrivcvgmul.5 |- ( ph -> E. m e. Z E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) )
ntrivcvgmul.6 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
ntrivcvgmul.7 |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
Assertion
Ref Expression
ntrivcvgmul |- ( ph -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) )
Distinct variable groups:   m, F, z   n, G, y   m, H, n, y, z, p   ph, m   w, m, y, z   n, p   ph, n   w, n, y, z, p   ph, y, z   y, w, z   m, Z, n, y, z   w, F   w, G   H, p, w   Z, p   k, F   k, G   k, H, m, n   ph, k, y, z   k, Z
Allowed substitution hints:   ph( w, p)   F( y, n, p)   G( z, m, p)   M( y, z, w, k, m, n, p)   Z( w)
Proof of Theorem ntrivcvgmul
StepHypRef Expression
1 ntrivcvgmul.3 . . 3 |- ( ph -> E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) )
2 ntrivcvgmul.5 . . 3 |- ( ph -> E. m e. Z E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) )
3 eeanv 2182 . . . . 5 |- ( E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) <-> ( E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) )
432rexbii 3042 . . . 4 |- ( E. n e. Z E. m e. Z E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) <-> E. n e. Z E. m e. Z ( E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) )
5 reeanv 3107 . . . 4 |- ( E. n e. Z E. m e. Z ( E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) <-> ( E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ E. m e. Z E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) )
64, 5bitri 264 . . 3 |- ( E. n e. Z E. m e. Z E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) <-> ( E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ E. m e. Z E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) )
71, 2, 6sylanbrc 698 . 2 |- ( ph -> E. n e. Z E. m e. Z E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) )
8 ntrivcvgmul.1 . . . . . . . . 9 |- Z = ( ZZ>= ` M )
9 uzssz 11707 . . . . . . . . 9 |- ( ZZ>= ` M ) C_ ZZ
108, 9eqsstri 3635 . . . . . . . 8 |- Z C_ ZZ
11 simp2l 1087 . . . . . . . 8 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> n e. Z )
1210, 11sseldi 3601 . . . . . . 7 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> n e. ZZ )
1312zred 11482 . . . . . 6 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> n e. RR )
14 simp2r 1088 . . . . . . . 8 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> m e. Z )
1510, 14sseldi 3601 . . . . . . 7 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> m e. ZZ )
1615zred 11482 . . . . . 6 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> m e. RR )
17 simpl2l 1114 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> n e. Z )
18 simpl2r 1115 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> m e. Z )
19 simp3ll 1132 . . . . . . . 8 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> y =/= 0 )
2019adantr 481 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> y =/= 0 )
21 simp3rl 1134 . . . . . . . 8 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> z =/= 0 )
2221adantr 481 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> z =/= 0 )
23 simp3lr 1133 . . . . . . . 8 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> seq n ( x. , F ) ~~> y )
2423adantr 481 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> seq n ( x. , F ) ~~> y )
25 simp3rr 1135 . . . . . . . 8 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> seq m ( x. , G ) ~~> z )
2625adantr 481 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> seq m ( x. , G ) ~~> z )
27 simpl1 1064 . . . . . . . 8 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> ph )
28 ntrivcvgmul.4 . . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
2927, 28sylan 488 . . . . . . 7 |- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) /\ k e. Z ) -> ( F ` k ) e. CC )
30 ntrivcvgmul.6 . . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
3127, 30sylan 488 . . . . . . 7 |- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) /\ k e. Z ) -> ( G ` k ) e. CC )
32 simpr 477 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> n <_ m )
33 ntrivcvgmul.7 . . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
3427, 33sylan 488 . . . . . . 7 |- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) /\ k e. Z ) -> ( H ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
358, 17, 18, 20, 22, 24, 26, 29, 31, 32, 34ntrivcvgmullem 14633 . . . . . 6 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ n <_ m ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) )
36 simpl2r 1115 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> m e. Z )
37 simpl2l 1114 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> n e. Z )
3821adantr 481 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> z =/= 0 )
3919adantr 481 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> y =/= 0 )
4025adantr 481 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> seq m ( x. , G ) ~~> z )
4123adantr 481 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> seq n ( x. , F ) ~~> y )
42 simpl1 1064 . . . . . . . 8 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> ph )
4342, 30sylan 488 . . . . . . 7 |- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) /\ k e. Z ) -> ( G ` k ) e. CC )
4442, 28sylan 488 . . . . . . 7 |- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) /\ k e. Z ) -> ( F ` k ) e. CC )
45 simpr 477 . . . . . . 7 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> m <_ n )
4628, 30mulcomd 10061 . . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) x. ( G ` k ) ) = ( ( G ` k ) x. ( F ` k ) ) )
4733, 46eqtrd 2656 . . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( H ` k ) = ( ( G ` k ) x. ( F ` k ) ) )
4842, 47sylan 488 . . . . . . 7 |- ( ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) /\ k e. Z ) -> ( H ` k ) = ( ( G ` k ) x. ( F ` k ) ) )
498, 36, 37, 38, 39, 40, 41, 43, 44, 45, 48ntrivcvgmullem 14633 . . . . . 6 |- ( ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) /\ m <_ n ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) )
5013, 16, 35, 49lecasei 10143 . . . . 5 |- ( ( ph /\ ( n e. Z /\ m e. Z ) /\ ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) )
51503expia 1267 . . . 4 |- ( ( ph /\ ( n e. Z /\ m e. Z ) ) -> ( ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) ) )
5251exlimdvv 1862 . . 3 |- ( ( ph /\ ( n e. Z /\ m e. Z ) ) -> ( E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) ) )
5352rexlimdvva 3038 . 2 |- ( ph -> ( E. n e. Z E. m e. Z E. y E. z ( ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) /\ ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) ) -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) ) )
547, 53mpd 15 1 |- ( ph -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , H ) ~~> w ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   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   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:  iprodmul  14734
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