![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fprodcllemf | Structured version Visualization version GIF version |
Description: Finite product closure lemma. A version of fprodcllem 14725 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fprodcllemf.ph | ⊢ Ⅎ𝑘𝜑 |
fprodcllemf.s | ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
fprodcllemf.xy | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) |
fprodcllemf.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodcllemf.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
fprodcllemf.1 | ⊢ (𝜑 → 1 ∈ 𝑆) |
Ref | Expression |
---|---|
fprodcllemf | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1a 3575 | . . 3 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
2 | nfcv 2793 | . . 3 ⊢ Ⅎ𝑗𝐴 | |
3 | nfcv 2793 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
4 | nfcv 2793 | . . 3 ⊢ Ⅎ𝑗𝐵 | |
5 | nfcsb1v 3582 | . . 3 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 | |
6 | 1, 2, 3, 4, 5 | cbvprod 14689 | . 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 |
7 | fprodcllemf.s | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
8 | fprodcllemf.xy | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) | |
9 | fprodcllemf.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
10 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ 𝐴) | |
11 | fprodcllemf.ph | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
12 | fprodcllemf.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) | |
13 | 12 | ex 449 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ 𝑆)) |
14 | 11, 13 | ralrimi 2986 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
15 | 14 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
16 | rspsbc 3551 | . . . . 5 ⊢ (𝑗 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → [𝑗 / 𝑘]𝐵 ∈ 𝑆)) | |
17 | 10, 15, 16 | sylc 65 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → [𝑗 / 𝑘]𝐵 ∈ 𝑆) |
18 | vex 3234 | . . . . 5 ⊢ 𝑗 ∈ V | |
19 | sbcel1g 4020 | . . . . 5 ⊢ (𝑗 ∈ V → ([𝑗 / 𝑘]𝐵 ∈ 𝑆 ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑆)) | |
20 | 18, 19 | ax-mp 5 | . . . 4 ⊢ ([𝑗 / 𝑘]𝐵 ∈ 𝑆 ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑆) |
21 | 17, 20 | sylib 208 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑆) |
22 | fprodcllemf.1 | . . 3 ⊢ (𝜑 → 1 ∈ 𝑆) | |
23 | 7, 8, 9, 21, 22 | fprodcllem 14725 | . 2 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑆) |
24 | 6, 23 | syl5eqel 2734 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 Ⅎwnf 1748 ∈ wcel 2030 ∀wral 2941 Vcvv 3231 [wsbc 3468 ⦋csb 3566 ⊆ wss 3607 (class class class)co 6690 Fincfn 7997 ℂcc 9972 1c1 9975 · cmul 9979 ∏cprod 14679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-fz 12365 df-fzo 12505 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-prod 14680 |
This theorem is referenced by: fprodreclf 14733 fprodn0f 14766 fprodclf 14767 fprodge0 14768 fprodge1 14770 |
Copyright terms: Public domain | W3C validator |