Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fprodcllem | Structured version Visualization version GIF version |
Description: Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) |
Ref | Expression |
---|---|
fprodcllem.1 | ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
fprodcllem.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) |
fprodcllem.3 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodcllem.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
fprodcllem.5 | ⊢ (𝜑 → 1 ∈ 𝑆) |
Ref | Expression |
---|---|
fprodcllem | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodeq1 15257 | . . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵) | |
2 | prod0 15291 | . . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1 | |
3 | 1, 2 | syl6eq 2872 | . . . 4 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = 1) |
4 | 3 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = 1) |
5 | fprodcllem.5 | . . . 4 ⊢ (𝜑 → 1 ∈ 𝑆) | |
6 | 5 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → 1 ∈ 𝑆) |
7 | 4, 6 | eqeltrd 2913 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
8 | fprodcllem.1 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝑆 ⊆ ℂ) |
10 | fprodcllem.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) | |
11 | 10 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) |
12 | fprodcllem.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
13 | 12 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Fin) |
14 | fprodcllem.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) | |
15 | 14 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
16 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
17 | 9, 11, 13, 15, 16 | fprodcl2lem 15298 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
18 | 7, 17 | pm2.61dane 3104 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3936 ∅c0 4291 (class class class)co 7150 Fincfn 8503 ℂcc 10529 1c1 10532 · cmul 10536 ∏cprod 15253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-prod 15254 |
This theorem is referenced by: fprodcl 15300 fprodrecl 15301 fprodzcl 15302 fprodnncl 15303 fprodrpcl 15304 fprodnn0cl 15305 fprodcllemf 15306 risefaccllem 15361 fallfaccllem 15362 |
Copyright terms: Public domain | W3C validator |