Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fprodclf | Structured version Visualization version GIF version |
Description: Closure of a finite product of complex numbers. A version of fprodcl 15660 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fprodclf.kph | ⊢ Ⅎ𝑘𝜑 |
fprodclf.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodclf.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
fprodclf | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprodclf.kph | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | ssidd 3945 | . 2 ⊢ (𝜑 → ℂ ⊆ ℂ) | |
3 | mulcl 10953 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
4 | 3 | adantl 482 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ) |
5 | fprodclf.a | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
6 | fprodclf.b | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
7 | 1cnd 10968 | . 2 ⊢ (𝜑 → 1 ∈ ℂ) | |
8 | 1, 2, 4, 5, 6, 7 | fprodcllemf 15666 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 Ⅎwnf 1786 ∈ wcel 2106 (class class class)co 7277 Fincfn 8731 ℂcc 10867 · cmul 10874 ∏cprod 15613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-inf2 9397 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-pre-sup 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-isom 6444 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-sup 9199 df-oi 9267 df-card 9695 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-n0 12232 df-z 12318 df-uz 12581 df-rp 12729 df-fz 13238 df-fzo 13381 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-prod 15614 |
This theorem is referenced by: fprodeq0g 15702 fprodle 15704 fprodexp 43105 fprodabs2 43106 fprod0 43107 mccllem 43108 fprodsubrecnncnvlem 43418 fprodaddrecnncnvlem 43420 dvmptfprodlem 43455 dvnprodlem2 43458 |
Copyright terms: Public domain | W3C validator |