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Mirrors > Home > MPE Home > Th. List > fprodcllemf | Structured version Visualization version GIF version |
Description: Finite product closure lemma. A version of fprodcllem 15984 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 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑗𝐵 | |
2 | nfcsb1v 3933 | . . 3 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 | |
3 | csbeq1a 3922 | . . 3 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
4 | 1, 2, 3 | cbvprodi 15948 | . 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 |
5 | fprodcllemf.s | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
6 | fprodcllemf.xy | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) | |
7 | fprodcllemf.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
8 | fprodcllemf.ph | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
9 | fprodcllemf.b | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) | |
10 | 9 | ex 412 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ 𝑆)) |
11 | 8, 10 | ralrimi 3255 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
12 | rspsbc 3888 | . . . . 5 ⊢ (𝑗 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → [𝑗 / 𝑘]𝐵 ∈ 𝑆)) | |
13 | 11, 12 | mpan9 506 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → [𝑗 / 𝑘]𝐵 ∈ 𝑆) |
14 | sbcel1g 4422 | . . . . 5 ⊢ (𝑗 ∈ V → ([𝑗 / 𝑘]𝐵 ∈ 𝑆 ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑆)) | |
15 | 14 | elv 3483 | . . . 4 ⊢ ([𝑗 / 𝑘]𝐵 ∈ 𝑆 ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑆) |
16 | 13, 15 | sylib 218 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑆) |
17 | fprodcllemf.1 | . . 3 ⊢ (𝜑 → 1 ∈ 𝑆) | |
18 | 5, 6, 7, 16, 17 | fprodcllem 15984 | . 2 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 ∈ 𝑆) |
19 | 4, 18 | eqeltrid 2843 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 Ⅎwnf 1780 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 [wsbc 3791 ⦋csb 3908 ⊆ wss 3963 (class class class)co 7431 Fincfn 8984 ℂcc 11151 1c1 11154 · cmul 11158 ∏cprod 15936 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-prod 15937 |
This theorem is referenced by: fprodreclf 15992 fprodn0f 16024 fprodclf 16025 fprodge0 16026 fprodge1 16028 |
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