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Mirrors > Home > MPE Home > Th. List > fprodconst | Structured version Visualization version GIF version |
Description: The product of constant terms (𝑘 is not free in 𝐵.) (Contributed by Scott Fenton, 12-Jan-2018.) |
Ref | Expression |
---|---|
fprodconst | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ 𝐴 𝐵 = (𝐵↑(♯‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exp0 13071 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (𝐵↑0) = 1) | |
2 | 1 | eqcomd 2777 | . . . 4 ⊢ (𝐵 ∈ ℂ → 1 = (𝐵↑0)) |
3 | prodeq1 14846 | . . . . . 6 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵) | |
4 | prod0 14880 | . . . . . 6 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1 | |
5 | 3, 4 | syl6eq 2821 | . . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = 1) |
6 | fveq2 6332 | . . . . . . 7 ⊢ (𝐴 = ∅ → (♯‘𝐴) = (♯‘∅)) | |
7 | hash0 13360 | . . . . . . 7 ⊢ (♯‘∅) = 0 | |
8 | 6, 7 | syl6eq 2821 | . . . . . 6 ⊢ (𝐴 = ∅ → (♯‘𝐴) = 0) |
9 | 8 | oveq2d 6809 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐵↑(♯‘𝐴)) = (𝐵↑0)) |
10 | 5, 9 | eqeq12d 2786 | . . . 4 ⊢ (𝐴 = ∅ → (∏𝑘 ∈ 𝐴 𝐵 = (𝐵↑(♯‘𝐴)) ↔ 1 = (𝐵↑0))) |
11 | 2, 10 | syl5ibrcom 237 | . . 3 ⊢ (𝐵 ∈ ℂ → (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = (𝐵↑(♯‘𝐴)))) |
12 | 11 | adantl 467 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = (𝐵↑(♯‘𝐴)))) |
13 | eqidd 2772 | . . . . . . 7 ⊢ (𝑘 = (𝑓‘𝑛) → 𝐵 = 𝐵) | |
14 | simprl 754 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ) | |
15 | simprr 756 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) | |
16 | simpllr 760 | . . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
17 | simpllr 760 | . . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝐵 ∈ ℂ) | |
18 | elfznn 12577 | . . . . . . . . 9 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ) | |
19 | 18 | adantl 467 | . . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑛 ∈ ℕ) |
20 | fvconst2g 6611 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝑛 ∈ ℕ) → ((ℕ × {𝐵})‘𝑛) = 𝐵) | |
21 | 17, 19, 20 | syl2anc 573 | . . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((ℕ × {𝐵})‘𝑛) = 𝐵) |
22 | 13, 14, 15, 16, 21 | fprod 14878 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑘 ∈ 𝐴 𝐵 = (seq1( · , (ℕ × {𝐵}))‘(♯‘𝐴))) |
23 | expnnval 13070 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ (♯‘𝐴) ∈ ℕ) → (𝐵↑(♯‘𝐴)) = (seq1( · , (ℕ × {𝐵}))‘(♯‘𝐴))) | |
24 | 23 | ad2ant2lr 742 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐵↑(♯‘𝐴)) = (seq1( · , (ℕ × {𝐵}))‘(♯‘𝐴))) |
25 | 22, 24 | eqtr4d 2808 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑘 ∈ 𝐴 𝐵 = (𝐵↑(♯‘𝐴))) |
26 | 25 | expr 444 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐵↑(♯‘𝐴)))) |
27 | 26 | exlimdv 2013 | . . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐵↑(♯‘𝐴)))) |
28 | 27 | expimpd 441 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 𝐵 = (𝐵↑(♯‘𝐴)))) |
29 | fz1f1o 14649 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) | |
30 | 29 | adantr 466 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) |
31 | 12, 28, 30 | mpjaod 847 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ 𝐴 𝐵 = (𝐵↑(♯‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∨ wo 834 = wceq 1631 ∃wex 1852 ∈ wcel 2145 ∅c0 4063 {csn 4316 × cxp 5247 –1-1-onto→wf1o 6030 ‘cfv 6031 (class class class)co 6793 Fincfn 8109 ℂcc 10136 0cc0 10138 1c1 10139 · cmul 10143 ℕcn 11222 ...cfz 12533 seqcseq 13008 ↑cexp 13067 ♯chash 13321 ∏cprod 14842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11495 df-z 11580 df-uz 11889 df-rp 12036 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-prod 14843 |
This theorem is referenced by: risefallfac 14961 gausslemma2dlem5 25317 gausslemma2dlem6 25318 breprexpnat 31052 circlemethnat 31059 circlevma 31060 circlemethhgt 31061 bcprod 31962 etransclem23 40991 hoicvrrex 41290 ovnhoilem1 41335 vonsn 41425 |
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