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Theorem prodfap0 11570
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 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
prodfap0.2 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
prodfap0.3 ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) # 0)
Assertion
Ref Expression
prodfap0 (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) # 0)
Distinct variable groups:   π‘˜,𝐹   π‘˜,𝑀   π‘˜,𝑁   πœ‘,π‘˜
Proof of Theorem prodfap0
Dummy variables 𝑛 𝑣 π‘š are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prodfap0.1 . . 3 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
2 eluzfz2 10049 . . 3 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (πœ‘ β†’ 𝑁 ∈ (𝑀...𝑁))
4 fveq2 5529 . . . . 5 (π‘š = 𝑀 β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = (seq𝑀( Β· , 𝐹)β€˜π‘€))
54breq1d 4027 . . . 4 (π‘š = 𝑀 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) # 0 ↔ (seq𝑀( Β· , 𝐹)β€˜π‘€) # 0))
65imbi2d 230 . . 3 (π‘š = 𝑀 β†’ ((πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) # 0) ↔ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘€) # 0)))
7 fveq2 5529 . . . . 5 (π‘š = 𝑛 β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = (seq𝑀( Β· , 𝐹)β€˜π‘›))
87breq1d 4027 . . . 4 (π‘š = 𝑛 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) # 0 ↔ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0))
98imbi2d 230 . . 3 (π‘š = 𝑛 β†’ ((πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) # 0) ↔ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0)))
10 fveq2 5529 . . . . 5 (π‘š = (𝑛 + 1) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)))
1110breq1d 4027 . . . 4 (π‘š = (𝑛 + 1) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) # 0 ↔ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) # 0))
1211imbi2d 230 . . 3 (π‘š = (𝑛 + 1) β†’ ((πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) # 0) ↔ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) # 0)))
13 fveq2 5529 . . . . 5 (π‘š = 𝑁 β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = (seq𝑀( Β· , 𝐹)β€˜π‘))
1413breq1d 4027 . . . 4 (π‘š = 𝑁 β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘š) # 0 ↔ (seq𝑀( Β· , 𝐹)β€˜π‘) # 0))
1514imbi2d 230 . . 3 (π‘š = 𝑁 β†’ ((πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) # 0) ↔ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) # 0)))
16 eluzfz1 10048 . . . 4 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑀 ∈ (𝑀...𝑁))
17 elfzelz 10042 . . . . . . . 8 (𝑀 ∈ (𝑀...𝑁) β†’ 𝑀 ∈ β„€)
1817adantl 277 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (𝑀...𝑁)) β†’ 𝑀 ∈ β„€)
19 prodfap0.2 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
2019adantlr 477 . . . . . . 7 (((πœ‘ ∧ 𝑀 ∈ (𝑀...𝑁)) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
21 mulcl 7955 . . . . . . . 8 ((π‘˜ ∈ β„‚ ∧ 𝑣 ∈ β„‚) β†’ (π‘˜ Β· 𝑣) ∈ β„‚)
2221adantl 277 . . . . . . 7 (((πœ‘ ∧ 𝑀 ∈ (𝑀...𝑁)) ∧ (π‘˜ ∈ β„‚ ∧ 𝑣 ∈ β„‚)) β†’ (π‘˜ Β· 𝑣) ∈ β„‚)
2318, 20, 22seq3-1 10477 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (𝑀...𝑁)) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘€) = (πΉβ€˜π‘€))
24 fveq2 5529 . . . . . . . . . 10 (π‘˜ = 𝑀 β†’ (πΉβ€˜π‘˜) = (πΉβ€˜π‘€))
2524breq1d 4027 . . . . . . . . 9 (π‘˜ = 𝑀 β†’ ((πΉβ€˜π‘˜) # 0 ↔ (πΉβ€˜π‘€) # 0))
2625imbi2d 230 . . . . . . . 8 (π‘˜ = 𝑀 β†’ ((πœ‘ β†’ (πΉβ€˜π‘˜) # 0) ↔ (πœ‘ β†’ (πΉβ€˜π‘€) # 0)))
27 prodfap0.3 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) # 0)
2827expcom 116 . . . . . . . 8 (π‘˜ ∈ (𝑀...𝑁) β†’ (πœ‘ β†’ (πΉβ€˜π‘˜) # 0))
2926, 28vtoclga 2817 . . . . . . 7 (𝑀 ∈ (𝑀...𝑁) β†’ (πœ‘ β†’ (πΉβ€˜π‘€) # 0))
3029impcom 125 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘€) # 0)
3123, 30eqbrtrd 4039 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (𝑀...𝑁)) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘€) # 0)
3231expcom 116 . . . 4 (𝑀 ∈ (𝑀...𝑁) β†’ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘€) # 0))
3316, 32syl 14 . . 3 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘€) # 0))
34 elfzouz 10168 . . . . . . . . 9 (𝑛 ∈ (𝑀..^𝑁) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
35343ad2ant2 1020 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
36193ad2antl1 1160 . . . . . . . 8 (((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)
3721adantl 277 . . . . . . . 8 (((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) ∧ (π‘˜ ∈ β„‚ ∧ 𝑣 ∈ β„‚)) β†’ (π‘˜ Β· 𝑣) ∈ β„‚)
3835, 36, 37seq3p1 10479 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) = ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))))
39 elfzofz 10179 . . . . . . . . . 10 (𝑛 ∈ (𝑀..^𝑁) β†’ 𝑛 ∈ (𝑀...𝑁))
40 elfzuz 10038 . . . . . . . . . . 11 (𝑛 ∈ (𝑀...𝑁) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
41 eqid 2188 . . . . . . . . . . . . 13 (β„€β‰₯β€˜π‘€) = (β„€β‰₯β€˜π‘€)
421, 16, 173syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑀 ∈ β„€)
4341, 42, 19prodf 11563 . . . . . . . . . . . 12 (πœ‘ β†’ seq𝑀( Β· , 𝐹):(β„€β‰₯β€˜π‘€)βŸΆβ„‚)
4443ffvelcdmda 5666 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) ∈ β„‚)
4540, 44sylan2 286 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (𝑀...𝑁)) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) ∈ β„‚)
4639, 45sylan2 286 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁)) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) ∈ β„‚)
47463adant3 1018 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) ∈ β„‚)
48 fzofzp1 10244 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) β†’ (𝑛 + 1) ∈ (𝑀...𝑁))
49 fveq2 5529 . . . . . . . . . . . . . 14 (π‘˜ = (𝑛 + 1) β†’ (πΉβ€˜π‘˜) = (πΉβ€˜(𝑛 + 1)))
5049eleq1d 2257 . . . . . . . . . . . . 13 (π‘˜ = (𝑛 + 1) β†’ ((πΉβ€˜π‘˜) ∈ β„‚ ↔ (πΉβ€˜(𝑛 + 1)) ∈ β„‚))
5150imbi2d 230 . . . . . . . . . . . 12 (π‘˜ = (𝑛 + 1) β†’ ((πœ‘ β†’ (πΉβ€˜π‘˜) ∈ β„‚) ↔ (πœ‘ β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)))
52 elfzuz 10038 . . . . . . . . . . . . 13 (π‘˜ ∈ (𝑀...𝑁) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘€))
5319expcom 116 . . . . . . . . . . . . 13 (π‘˜ ∈ (β„€β‰₯β€˜π‘€) β†’ (πœ‘ β†’ (πΉβ€˜π‘˜) ∈ β„‚))
5452, 53syl 14 . . . . . . . . . . . 12 (π‘˜ ∈ (𝑀...𝑁) β†’ (πœ‘ β†’ (πΉβ€˜π‘˜) ∈ β„‚))
5551, 54vtoclga 2817 . . . . . . . . . . 11 ((𝑛 + 1) ∈ (𝑀...𝑁) β†’ (πœ‘ β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚))
5648, 55syl 14 . . . . . . . . . 10 (𝑛 ∈ (𝑀..^𝑁) β†’ (πœ‘ β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚))
5756impcom 125 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁)) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
58573adant3 1018 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
59 simp3 1000 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0)
6049breq1d 4027 . . . . . . . . . . . . 13 (π‘˜ = (𝑛 + 1) β†’ ((πΉβ€˜π‘˜) # 0 ↔ (πΉβ€˜(𝑛 + 1)) # 0))
6160imbi2d 230 . . . . . . . . . . . 12 (π‘˜ = (𝑛 + 1) β†’ ((πœ‘ β†’ (πΉβ€˜π‘˜) # 0) ↔ (πœ‘ β†’ (πΉβ€˜(𝑛 + 1)) # 0)))
6261, 28vtoclga 2817 . . . . . . . . . . 11 ((𝑛 + 1) ∈ (𝑀...𝑁) β†’ (πœ‘ β†’ (πΉβ€˜(𝑛 + 1)) # 0))
6362impcom 125 . . . . . . . . . 10 ((πœ‘ ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) β†’ (πΉβ€˜(𝑛 + 1)) # 0)
6448, 63sylan2 286 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁)) β†’ (πΉβ€˜(𝑛 + 1)) # 0)
65643adant3 1018 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (πΉβ€˜(𝑛 + 1)) # 0)
6647, 58, 59, 65mulap0d 8632 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) Β· (πΉβ€˜(𝑛 + 1))) # 0)
6738, 66eqbrtrd 4039 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) # 0)
68673exp 1203 . . . . 5 (πœ‘ β†’ (𝑛 ∈ (𝑀..^𝑁) β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) # 0 β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) # 0)))
6968com12 30 . . . 4 (𝑛 ∈ (𝑀..^𝑁) β†’ (πœ‘ β†’ ((seq𝑀( Β· , 𝐹)β€˜π‘›) # 0 β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) # 0)))
7069a2d 26 . . 3 (𝑛 ∈ (𝑀..^𝑁) β†’ ((πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) # 0)))
716, 9, 12, 15, 33, 70fzind2 10256 . 2 (𝑁 ∈ (𝑀...𝑁) β†’ (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) # 0))
723, 71mpcom 36 1 (πœ‘ β†’ (seq𝑀( Β· , 𝐹)β€˜π‘) # 0)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ∧ w3a 979   = wceq 1363   ∈ wcel 2159   class class class wbr 4017  β€˜cfv 5230  (class class class)co 5890  β„‚cc 7826  0cc0 7828  1c1 7829   + caddc 7831   Β· cmul 7833   # cap 8555  β„€cz 9270  β„€β‰₯cuz 9545  ...cfz 10025  ..^cfzo 10159  seqcseq 10462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-inn 8937  df-n0 9194  df-z 9271  df-uz 9546  df-fz 10026  df-fzo 10160  df-seqfrec 10463
This theorem is referenced by:  prodfrecap  11571  prodfdivap  11572
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