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Theorem prodfap0 11453
Description: The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.)
Hypotheses
Ref Expression
prodfap0.1 (𝜑𝑁 ∈ (ℤ𝑀))
prodfap0.2 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)
prodfap0.3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) # 0)
Assertion
Ref Expression
prodfap0 (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) # 0)
Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑁   𝜑,𝑘

Proof of Theorem prodfap0
Dummy variables 𝑛 𝑣 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prodfap0.1 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 9940 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 fveq2 5470 . . . . 5 (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀))
54breq1d 3977 . . . 4 (𝑚 = 𝑀 → ((seq𝑀( · , 𝐹)‘𝑚) # 0 ↔ (seq𝑀( · , 𝐹)‘𝑀) # 0))
65imbi2d 229 . . 3 (𝑚 = 𝑀 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) # 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) # 0)))
7 fveq2 5470 . . . . 5 (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛))
87breq1d 3977 . . . 4 (𝑚 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑚) # 0 ↔ (seq𝑀( · , 𝐹)‘𝑛) # 0))
98imbi2d 229 . . 3 (𝑚 = 𝑛 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) # 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑛) # 0)))
10 fveq2 5470 . . . . 5 (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1)))
1110breq1d 3977 . . . 4 (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑚) # 0 ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0))
1211imbi2d 229 . . 3 (𝑚 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) # 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)))
13 fveq2 5470 . . . . 5 (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁))
1413breq1d 3977 . . . 4 (𝑚 = 𝑁 → ((seq𝑀( · , 𝐹)‘𝑚) # 0 ↔ (seq𝑀( · , 𝐹)‘𝑁) # 0))
1514imbi2d 229 . . 3 (𝑚 = 𝑁 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) # 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) # 0)))
16 eluzfz1 9939 . . . 4 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
17 elfzelz 9934 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
1817adantl 275 . . . . . . 7 ((𝜑𝑀 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
19 prodfap0.2 . . . . . . . 8 ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)
2019adantlr 469 . . . . . . 7 (((𝜑𝑀 ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)
21 mulcl 7861 . . . . . . . 8 ((𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑘 · 𝑣) ∈ ℂ)
2221adantl 275 . . . . . . 7 (((𝜑𝑀 ∈ (𝑀...𝑁)) ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 · 𝑣) ∈ ℂ)
2318, 20, 22seq3-1 10368 . . . . . 6 ((𝜑𝑀 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹𝑀))
24 fveq2 5470 . . . . . . . . . 10 (𝑘 = 𝑀 → (𝐹𝑘) = (𝐹𝑀))
2524breq1d 3977 . . . . . . . . 9 (𝑘 = 𝑀 → ((𝐹𝑘) # 0 ↔ (𝐹𝑀) # 0))
2625imbi2d 229 . . . . . . . 8 (𝑘 = 𝑀 → ((𝜑 → (𝐹𝑘) # 0) ↔ (𝜑 → (𝐹𝑀) # 0)))
27 prodfap0.3 . . . . . . . . 9 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) # 0)
2827expcom 115 . . . . . . . 8 (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹𝑘) # 0))
2926, 28vtoclga 2778 . . . . . . 7 (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐹𝑀) # 0))
3029impcom 124 . . . . . 6 ((𝜑𝑀 ∈ (𝑀...𝑁)) → (𝐹𝑀) # 0)
3123, 30eqbrtrd 3988 . . . . 5 ((𝜑𝑀 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑀) # 0)
3231expcom 115 . . . 4 (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) # 0))
3316, 32syl 14 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) # 0))
34 elfzouz 10059 . . . . . . . . 9 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ𝑀))
35343ad2ant2 1004 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → 𝑛 ∈ (ℤ𝑀))
36193ad2antl1 1144 . . . . . . . 8 (((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) ∧ 𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)
3721adantl 275 . . . . . . . 8 (((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) ∧ (𝑘 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑘 · 𝑣) ∈ ℂ)
3835, 36, 37seq3p1 10370 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))))
39 elfzofz 10070 . . . . . . . . . 10 (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (𝑀...𝑁))
40 elfzuz 9930 . . . . . . . . . . 11 (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ𝑀))
41 eqid 2157 . . . . . . . . . . . . 13 (ℤ𝑀) = (ℤ𝑀)
421, 16, 173syl 17 . . . . . . . . . . . . 13 (𝜑𝑀 ∈ ℤ)
4341, 42, 19prodf 11446 . . . . . . . . . . . 12 (𝜑 → seq𝑀( · , 𝐹):(ℤ𝑀)⟶ℂ)
4443ffvelrnda 5604 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑀)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
4540, 44sylan2 284 . . . . . . . . . 10 ((𝜑𝑛 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
4639, 45sylan2 284 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
47463adant3 1002 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ)
48 fzofzp1 10135 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁))
49 fveq2 5470 . . . . . . . . . . . . . 14 (𝑘 = (𝑛 + 1) → (𝐹𝑘) = (𝐹‘(𝑛 + 1)))
5049eleq1d 2226 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ))
5150imbi2d 229 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹𝑘) ∈ ℂ) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ)))
52 elfzuz 9930 . . . . . . . . . . . . 13 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ𝑀))
5319expcom 115 . . . . . . . . . . . . 13 (𝑘 ∈ (ℤ𝑀) → (𝜑 → (𝐹𝑘) ∈ ℂ))
5452, 53syl 14 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹𝑘) ∈ ℂ))
5551, 54vtoclga 2778 . . . . . . . . . . 11 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
5648, 55syl 14 . . . . . . . . . 10 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))
5756impcom 124 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
58573adant3 1002 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (𝐹‘(𝑛 + 1)) ∈ ℂ)
59 simp3 984 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘𝑛) # 0)
6049breq1d 3977 . . . . . . . . . . . . 13 (𝑘 = (𝑛 + 1) → ((𝐹𝑘) # 0 ↔ (𝐹‘(𝑛 + 1)) # 0))
6160imbi2d 229 . . . . . . . . . . . 12 (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹𝑘) # 0) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) # 0)))
6261, 28vtoclga 2778 . . . . . . . . . . 11 ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) # 0))
6362impcom 124 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑛 + 1)) # 0)
6448, 63sylan2 284 . . . . . . . . 9 ((𝜑𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) # 0)
65643adant3 1002 . . . . . . . 8 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (𝐹‘(𝑛 + 1)) # 0)
6647, 58, 59, 65mulap0d 8536 . . . . . . 7 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) # 0)
6738, 66eqbrtrd 3988 . . . . . 6 ((𝜑𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) # 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)
68673exp 1184 . . . . 5 (𝜑 → (𝑛 ∈ (𝑀..^𝑁) → ((seq𝑀( · , 𝐹)‘𝑛) # 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)))
6968com12 30 . . . 4 (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) # 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)))
7069a2d 26 . . 3 (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑛) # 0) → (𝜑 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) # 0)))
716, 9, 12, 15, 33, 70fzind2 10147 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) # 0))
723, 71mpcom 36 1 (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) # 0)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1335  wcel 2128   class class class wbr 3967  cfv 5172  (class class class)co 5826  cc 7732  0cc0 7734  1c1 7735   + caddc 7737   · cmul 7739   # cap 8460  cz 9172  cuz 9444  ...cfz 9918  ..^cfzo 10050  seqcseq 10353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-inn 8839  df-n0 9096  df-z 9173  df-uz 9445  df-fz 9919  df-fzo 10051  df-seqfrec 10354
This theorem is referenced by:  prodfrecap  11454  prodfdivap  11455
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