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Theorem prodfap0 11552
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 10031 . . 3 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (πœ‘ β†’ 𝑁 ∈ (𝑀...𝑁))
4 fveq2 5515 . . . . 5 (π‘š = 𝑀 β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = (seq𝑀( Β· , 𝐹)β€˜π‘€))
54breq1d 4013 . . . 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 5515 . . . . 5 (π‘š = 𝑛 β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = (seq𝑀( Β· , 𝐹)β€˜π‘›))
87breq1d 4013 . . . 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 5515 . . . . 5 (π‘š = (𝑛 + 1) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)))
1110breq1d 4013 . . . 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 5515 . . . . 5 (π‘š = 𝑁 β†’ (seq𝑀( Β· , 𝐹)β€˜π‘š) = (seq𝑀( Β· , 𝐹)β€˜π‘))
1413breq1d 4013 . . . 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 10030 . . . 4 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑀 ∈ (𝑀...𝑁))
17 elfzelz 10024 . . . . . . . 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 7937 . . . . . . . 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 10459 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (𝑀...𝑁)) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘€) = (πΉβ€˜π‘€))
24 fveq2 5515 . . . . . . . . . 10 (π‘˜ = 𝑀 β†’ (πΉβ€˜π‘˜) = (πΉβ€˜π‘€))
2524breq1d 4013 . . . . . . . . 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 2803 . . . . . . 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 4025 . . . . 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 10150 . . . . . . . . 9 (𝑛 ∈ (𝑀..^𝑁) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
35343ad2ant2 1019 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
36193ad2antl1 1159 . . . . . . . 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 10461 . . . . . . 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 10161 . . . . . . . . . 10 (𝑛 ∈ (𝑀..^𝑁) β†’ 𝑛 ∈ (𝑀...𝑁))
40 elfzuz 10020 . . . . . . . . . . 11 (𝑛 ∈ (𝑀...𝑁) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘€))
41 eqid 2177 . . . . . . . . . . . . 13 (β„€β‰₯β€˜π‘€) = (β„€β‰₯β€˜π‘€)
421, 16, 173syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑀 ∈ β„€)
4341, 42, 19prodf 11545 . . . . . . . . . . . 12 (πœ‘ β†’ seq𝑀( Β· , 𝐹):(β„€β‰₯β€˜π‘€)βŸΆβ„‚)
4443ffvelcdmda 5651 . . . . . . . . . . 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 1017 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) ∈ β„‚)
48 fzofzp1 10226 . . . . . . . . . . 11 (𝑛 ∈ (𝑀..^𝑁) β†’ (𝑛 + 1) ∈ (𝑀...𝑁))
49 fveq2 5515 . . . . . . . . . . . . . 14 (π‘˜ = (𝑛 + 1) β†’ (πΉβ€˜π‘˜) = (πΉβ€˜(𝑛 + 1)))
5049eleq1d 2246 . . . . . . . . . . . . 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 10020 . . . . . . . . . . . . 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 2803 . . . . . . . . . . 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 1017 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (πΉβ€˜(𝑛 + 1)) ∈ β„‚)
59 simp3 999 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0)
6049breq1d 4013 . . . . . . . . . . . . 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 2803 . . . . . . . . . . 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 1017 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (πΉβ€˜(𝑛 + 1)) # 0)
6647, 58, 59, 65mulap0d 8614 . . . . . . 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 4025 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( Β· , 𝐹)β€˜π‘›) # 0) β†’ (seq𝑀( Β· , 𝐹)β€˜(𝑛 + 1)) # 0)
68673exp 1202 . . . . 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 10238 . 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 978   = wceq 1353   ∈ wcel 2148   class class class wbr 4003  β€˜cfv 5216  (class class class)co 5874  β„‚cc 7808  0cc0 7810  1c1 7811   + caddc 7813   Β· cmul 7815   # cap 8537  β„€cz 9252  β„€β‰₯cuz 9527  ...cfz 10007  ..^cfzo 10141  seqcseq 10444
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-inn 8919  df-n0 9176  df-z 9253  df-uz 9528  df-fz 10008  df-fzo 10142  df-seqfrec 10445
This theorem is referenced by:  prodfrecap  11553  prodfdivap  11554
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