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Theorem ntrivcvgfvn0 15849
Description: Any value of a product sequence that converges to a nonzero value is itself nonzero. (Contributed by Scott Fenton, 20-Dec-2017.)
Hypotheses
Ref Expression
ntrivcvgfvn0.1 𝑍 = (β„€β‰₯β€˜π‘€)
ntrivcvgfvn0.2 (πœ‘ β†’ 𝑁 ∈ 𝑍)
ntrivcvgfvn0.3 (πœ‘ β†’ seq𝑀( Β· , 𝐹) ⇝ 𝑋)
ntrivcvgfvn0.4 (πœ‘ β†’ 𝑋 β‰  0)
ntrivcvgfvn0.5 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
Assertion
Ref Expression
ntrivcvgfvn0 (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) β‰  0)
Distinct variable groups:   π‘˜,𝐹   πœ‘,π‘˜   π‘˜,𝑀   π‘˜,𝑁   π‘˜,𝑍
Allowed substitution hint:   𝑋(π‘˜)
Proof of Theorem ntrivcvgfvn0
Dummy variables π‘š 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ntrivcvgfvn0.4 . 2 (πœ‘ β†’ 𝑋 β‰  0)
2 fclim 15501 . . . . . . . 8 ⇝ :dom ⇝ βŸΆβ„‚
3 ffun 6719 . . . . . . . 8 ( ⇝ :dom ⇝ βŸΆβ„‚ β†’ Fun ⇝ )
42, 3ax-mp 5 . . . . . . 7 Fun ⇝
5 ntrivcvgfvn0.3 . . . . . . 7 (πœ‘ β†’ seq𝑀( Β· , 𝐹) ⇝ 𝑋)
6 funbrfv 6941 . . . . . . 7 (Fun ⇝ β†’ (seq𝑀( Β· , 𝐹) ⇝ 𝑋 β†’ ( ⇝ β€˜seq𝑀( Β· , 𝐹)) = 𝑋))
74, 5, 6mpsyl 68 . . . . . 6 (πœ‘ β†’ ( ⇝ β€˜seq𝑀( Β· , 𝐹)) = 𝑋)
87adantr 479 . . . . 5 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ ( ⇝ β€˜seq𝑀( Β· , 𝐹)) = 𝑋)
9 eqid 2730 . . . . . . 7 (β„€β‰₯β€˜π‘) = (β„€β‰₯β€˜π‘)
10 ntrivcvgfvn0.1 . . . . . . . . . 10 𝑍 = (β„€β‰₯β€˜π‘€)
11 uzssz 12847 . . . . . . . . . 10 (β„€β‰₯β€˜π‘€) βŠ† β„€
1210, 11eqsstri 4015 . . . . . . . . 9 𝑍 βŠ† β„€
13 ntrivcvgfvn0.2 . . . . . . . . 9 (πœ‘ β†’ 𝑁 ∈ 𝑍)
1412, 13sselid 3979 . . . . . . . 8 (πœ‘ β†’ 𝑁 ∈ β„€)
1514adantr 479 . . . . . . 7 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ 𝑁 ∈ β„€)
16 seqex 13972 . . . . . . . 8 seq𝑀( Β· , 𝐹) ∈ V
1716a1i 11 . . . . . . 7 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ seq𝑀( Β· , 𝐹) ∈ V)
18 0cnd 11211 . . . . . . 7 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ 0 ∈ β„‚)
19 fveqeq2 6899 . . . . . . . . . 10 (π‘š = 𝑁 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) = 0 ↔ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0))
2019imbi2d 339 . . . . . . . . 9 (π‘š = 𝑁 β†’ (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = 0) ↔ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0)))
21 fveqeq2 6899 . . . . . . . . . 10 (π‘š = 𝑛 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) = 0 ↔ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0))
2221imbi2d 339 . . . . . . . . 9 (π‘š = 𝑛 β†’ (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = 0) ↔ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0)))
23 fveqeq2 6899 . . . . . . . . . 10 (π‘š = (𝑛 + 1) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) = 0 ↔ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0))
2423imbi2d 339 . . . . . . . . 9 (π‘š = (𝑛 + 1) β†’ (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = 0) ↔ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0)))
25 fveqeq2 6899 . . . . . . . . . 10 (π‘š = π‘˜ β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) = 0 ↔ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = 0))
2625imbi2d 339 . . . . . . . . 9 (π‘š = π‘˜ β†’ (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = 0) ↔ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = 0)))
27 simpr 483 . . . . . . . . 9 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0)
2813, 10eleqtrdi 2841 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
29 uztrn 12844 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ 𝑁 ∈ (β„€β‰₯β€˜π‘€)) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
3028, 29sylan2 591 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
31303adant3 1130 . . . . . . . . . . . . . 14 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
32 seqp1 13985 . . . . . . . . . . . . . 14 (𝑛 ∈ (β„€β‰₯β€˜π‘€) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))))
3331, 32syl 17 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))))
34 oveq1 7418 . . . . . . . . . . . . . 14 ((seq𝑀( Β· , 𝐹)β€˜π‘›) = 0 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))) = (0 Β· (πΉβ€˜(𝑛 + 1))))
35343ad2ant3 1133 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))) = (0 Β· (πΉβ€˜(𝑛 + 1))))
36 peano2uz 12889 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ (𝑛 + 1) ∈ (β„€β‰₯β€˜π‘))
3710uztrn2 12845 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ 𝑍 ∧ (𝑛 + 1) ∈ (β„€β‰₯β€˜π‘)) β†’ (𝑛 + 1) ∈ 𝑍)
3813, 36, 37syl2an 594 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (𝑛 + 1) ∈ 𝑍)
39 ntrivcvgfvn0.5 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
4039ralrimiva 3144 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ βˆ€π‘˜ ∈ 𝑍 (πΉβ€˜π‘˜) ∈ β„‚)
41 fveq2 6890 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = (𝑛 + 1) β†’ (πΉβ€˜π‘˜) = (πΉβ€˜(𝑛 + 1)))
4241eleq1d 2816 . . . . . . . . . . . . . . . . . . 19 (π‘˜ = (𝑛 + 1) β†’ ((πΉβ€˜π‘˜) ∈ β„‚ ↔ (πΉβ€˜(𝑛 + 1)) ∈ β„‚))
4342rspcv 3607 . . . . . . . . . . . . . . . . . 18 ((𝑛 + 1) ∈ 𝑍 β†’ (βˆ€π‘˜ ∈ 𝑍 (πΉβ€˜π‘˜) ∈ β„‚ β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚))
4440, 43mpan9 505 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ (𝑛 + 1) ∈ 𝑍) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
4538, 44syldan 589 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
4645ancoms 457 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
4746mul02d 11416 . . . . . . . . . . . . . 14 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘) β†’ (0 Β· (πΉβ€˜(𝑛 + 1))) = 0)
48473adant3 1130 . . . . . . . . . . . . 13 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ (0 Β· (πΉβ€˜(𝑛 + 1))) = 0)
4933, 35, 483eqtrd 2774 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0)
50493exp 1117 . . . . . . . . . . 11 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ (πœ‘ β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) = 0 β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0)))
5150adantrd 490 . . . . . . . . . 10 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) = 0 β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0)))
5251a2d 29 . . . . . . . . 9 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) = 0) β†’ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = 0)))
5320, 22, 24, 26, 27, 52uzind4i 12898 . . . . . . . 8 (π‘˜ ∈ (β„€β‰₯β€˜π‘) β†’ ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = 0))
5453impcom 406 . . . . . . 7 (((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘˜) = 0)
559, 15, 17, 18, 54climconst 15491 . . . . . 6 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ seq𝑀( Β· , 𝐹) ⇝ 0)
56 funbrfv 6941 . . . . . 6 (Fun ⇝ β†’ (seq𝑀( Β· , 𝐹) ⇝ 0 β†’ ( ⇝ β€˜seq𝑀( Β· , 𝐹)) = 0))
574, 55, 56mpsyl 68 . . . . 5 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ ( ⇝ β€˜seq𝑀( Β· , 𝐹)) = 0)
588, 57eqtr3d 2772 . . . 4 ((πœ‘ ∧ (seq𝑀( Β· , 𝐹)β€˜π‘) = 0) β†’ 𝑋 = 0)
5958ex 411 . . 3 (πœ‘ β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘) = 0 β†’ 𝑋 = 0))
6059necon3d 2959 . 2 (πœ‘ β†’ (𝑋 β‰  0 β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) β‰  0))
611, 60mpd 15 1 (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) β‰  0)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  Vcvv 3472   class class class wbr 5147  dom cdm 5675  Fun wfun 6536  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  0cc0 11112  1c1 11113   + caddc 11115   Β· cmul 11117  β„€cz 12562  β„€β‰₯cuz 12826  seqcseq 13970   ⇝ cli 15432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436
This theorem is referenced by:  ntrivcvgtail  15850
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