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Mirrors > Home > MPE Home > Th. List > fproddivf | Structured version Visualization version GIF version |
Description: The quotient of two finite products. A version of fproddiv 15317 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fproddivf.kph | ⊢ Ⅎ𝑘𝜑 |
fproddivf.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fproddivf.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fproddivf.c | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
fproddivf.ne0 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ≠ 0) |
Ref | Expression |
---|---|
fproddivf | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵 / 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 / ∏𝑘 ∈ 𝐴 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑗(𝐵 / 𝐶) | |
2 | nfcsb1v 3909 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 | |
3 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑘 / | |
4 | nfcsb1v 3909 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐶 | |
5 | 2, 3, 4 | nfov 7188 | . . . 4 ⊢ Ⅎ𝑘(⦋𝑗 / 𝑘⦌𝐵 / ⦋𝑗 / 𝑘⦌𝐶) |
6 | csbeq1a 3899 | . . . . 5 ⊢ (𝑘 = 𝑗 → 𝐵 = ⦋𝑗 / 𝑘⦌𝐵) | |
7 | csbeq1a 3899 | . . . . 5 ⊢ (𝑘 = 𝑗 → 𝐶 = ⦋𝑗 / 𝑘⦌𝐶) | |
8 | 6, 7 | oveq12d 7176 | . . . 4 ⊢ (𝑘 = 𝑗 → (𝐵 / 𝐶) = (⦋𝑗 / 𝑘⦌𝐵 / ⦋𝑗 / 𝑘⦌𝐶)) |
9 | 1, 5, 8 | cbvprodi 15273 | . . 3 ⊢ ∏𝑘 ∈ 𝐴 (𝐵 / 𝐶) = ∏𝑗 ∈ 𝐴 (⦋𝑗 / 𝑘⦌𝐵 / ⦋𝑗 / 𝑘⦌𝐶) |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵 / 𝐶) = ∏𝑗 ∈ 𝐴 (⦋𝑗 / 𝑘⦌𝐵 / ⦋𝑗 / 𝑘⦌𝐶)) |
11 | fproddivf.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
12 | fproddivf.kph | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
13 | nfvd 1916 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑘 𝑗 ∈ 𝐴) | |
14 | 12, 13 | nfan1 2200 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝐴) |
15 | 2 | nfel1 2996 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ |
16 | 14, 15 | nfim 1897 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ) |
17 | eleq1w 2897 | . . . . . 6 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝐴 ↔ 𝑗 ∈ 𝐴)) | |
18 | 17 | anbi2d 630 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑗 ∈ 𝐴))) |
19 | 6 | eleq1d 2899 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐵 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ)) |
20 | 18, 19 | imbi12d 347 | . . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ))) |
21 | fproddivf.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
22 | 16, 20, 21 | chvarfv 2242 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐵 ∈ ℂ) |
23 | 4 | nfel1 2996 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ |
24 | 14, 23 | nfim 1897 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ) |
25 | 7 | eleq1d 2899 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐶 ∈ ℂ ↔ ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ)) |
26 | 18, 25 | imbi12d 347 | . . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ))) |
27 | fproddivf.c | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
28 | 24, 26, 27 | chvarfv 2242 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐶 ∈ ℂ) |
29 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑘0 | |
30 | 4, 29 | nfne 3121 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑗 / 𝑘⦌𝐶 ≠ 0 |
31 | 14, 30 | nfim 1897 | . . . 4 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐶 ≠ 0) |
32 | 7 | neeq1d 3077 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐶 ≠ 0 ↔ ⦋𝑗 / 𝑘⦌𝐶 ≠ 0)) |
33 | 18, 32 | imbi12d 347 | . . . 4 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ≠ 0) ↔ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐶 ≠ 0))) |
34 | fproddivf.ne0 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ≠ 0) | |
35 | 31, 33, 34 | chvarfv 2242 | . . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ⦋𝑗 / 𝑘⦌𝐶 ≠ 0) |
36 | 11, 22, 28, 35 | fproddiv 15317 | . 2 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 (⦋𝑗 / 𝑘⦌𝐵 / ⦋𝑗 / 𝑘⦌𝐶) = (∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 / ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶)) |
37 | nfcv 2979 | . . . . . 6 ⊢ Ⅎ𝑗𝐵 | |
38 | 37, 2, 6 | cbvprodi 15273 | . . . . 5 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 |
39 | 38 | eqcomi 2832 | . . . 4 ⊢ ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 = ∏𝑘 ∈ 𝐴 𝐵 |
40 | 39 | a1i 11 | . . 3 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 = ∏𝑘 ∈ 𝐴 𝐵) |
41 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑗𝐶 | |
42 | 7 | equcoms 2027 | . . . . . 6 ⊢ (𝑗 = 𝑘 → 𝐶 = ⦋𝑗 / 𝑘⦌𝐶) |
43 | 42 | eqcomd 2829 | . . . . 5 ⊢ (𝑗 = 𝑘 → ⦋𝑗 / 𝑘⦌𝐶 = 𝐶) |
44 | 4, 41, 43 | cbvprodi 15273 | . . . 4 ⊢ ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 = ∏𝑘 ∈ 𝐴 𝐶 |
45 | 44 | a1i 11 | . . 3 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶 = ∏𝑘 ∈ 𝐴 𝐶) |
46 | 40, 45 | oveq12d 7176 | . 2 ⊢ (𝜑 → (∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐵 / ∏𝑗 ∈ 𝐴 ⦋𝑗 / 𝑘⦌𝐶) = (∏𝑘 ∈ 𝐴 𝐵 / ∏𝑘 ∈ 𝐴 𝐶)) |
47 | 10, 36, 46 | 3eqtrd 2862 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵 / 𝐶) = (∏𝑘 ∈ 𝐴 𝐵 / ∏𝑘 ∈ 𝐴 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ≠ wne 3018 ⦋csb 3885 (class class class)co 7158 Fincfn 8511 ℂcc 10537 0cc0 10539 / cdiv 11299 ∏cprod 15261 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-prod 15262 |
This theorem is referenced by: fprodle 15352 |
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