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Mirrors > Home > MPE Home > Th. List > fprodsplit1f | Structured version Visualization version GIF version |
Description: Separate out a term in a finite product. A version of fprodsplit1 45549 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fprodsplit1f.kph | ⊢ Ⅎ𝑘𝜑 |
fprodsplit1f.fk | ⊢ (𝜑 → Ⅎ𝑘𝐷) |
fprodsplit1f.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodsplit1f.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fprodsplit1f.c | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
fprodsplit1f.d | ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
fprodsplit1f | ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprodsplit1f.kph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | disjdif 4478 | . . . 4 ⊢ ({𝐶} ∩ (𝐴 ∖ {𝐶})) = ∅ | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐶} ∩ (𝐴 ∖ {𝐶})) = ∅) |
4 | fprodsplit1f.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
5 | 4 | snssd 4814 | . . . . 5 ⊢ (𝜑 → {𝐶} ⊆ 𝐴) |
6 | undif 4488 | . . . . 5 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) | |
7 | 5, 6 | sylib 218 | . . . 4 ⊢ (𝜑 → ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) |
8 | 7 | eqcomd 2741 | . . 3 ⊢ (𝜑 → 𝐴 = ({𝐶} ∪ (𝐴 ∖ {𝐶}))) |
9 | fprodsplit1f.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
10 | fprodsplit1f.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
11 | 1, 3, 8, 9, 10 | fprodsplitf 16021 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (∏𝑘 ∈ {𝐶}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
12 | 4 | ancli 548 | . . . . . 6 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ 𝐴)) |
13 | nfv 1912 | . . . . . . . . 9 ⊢ Ⅎ𝑘 𝐶 ∈ 𝐴 | |
14 | 1, 13 | nfan 1897 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ 𝐴) |
15 | nfcsb1v 3933 | . . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 | |
16 | 15 | nfel1 2920 | . . . . . . . 8 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ |
17 | 14, 16 | nfim 1894 | . . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
18 | eleq1 2827 | . . . . . . . . 9 ⊢ (𝑘 = 𝐶 → (𝑘 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
19 | 18 | anbi2d 630 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐶 ∈ 𝐴))) |
20 | csbeq1a 3922 | . . . . . . . . 9 ⊢ (𝑘 = 𝐶 → 𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
21 | 20 | eleq1d 2824 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → (𝐵 ∈ ℂ ↔ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
22 | 19, 21 | imbi12d 344 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ))) |
23 | 17, 22, 10 | vtoclg1f 3570 | . . . . . 6 ⊢ (𝐶 ∈ 𝐴 → ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
24 | 4, 12, 23 | sylc 65 | . . . . 5 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
25 | prodsns 16005 | . . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝐶}𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
26 | 4, 24, 25 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {𝐶}𝐵 = ⦋𝐶 / 𝑘⦌𝐵) |
27 | fprodsplit1f.fk | . . . . 5 ⊢ (𝜑 → Ⅎ𝑘𝐷) | |
28 | fprodsplit1f.d | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷) | |
29 | 1, 27, 4, 28 | csbiedf 3939 | . . . 4 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 = 𝐷) |
30 | 26, 29 | eqtrd 2775 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐶}𝐵 = 𝐷) |
31 | 30 | oveq1d 7446 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐶}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵) = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
32 | 11, 31 | eqtrd 2775 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ⦋csb 3908 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {csn 4631 (class class class)co 7431 Fincfn 8984 ℂcc 11151 · 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: fprodeq0g 16027 fprodsplit1 45549 fprod0 45552 dvmptfprodlem 45900 |
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