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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 41881 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 4423 | . . . 4 ⊢ ({𝐶} ∩ (𝐴 ∖ {𝐶})) = ∅ | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐶} ∩ (𝐴 ∖ {𝐶})) = ∅) |
4 | fprodsplit1f.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
5 | 4 | snssd 4744 | . . . . 5 ⊢ (𝜑 → {𝐶} ⊆ 𝐴) |
6 | undif 4432 | . . . . 5 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) | |
7 | 5, 6 | sylib 220 | . . . 4 ⊢ (𝜑 → ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) |
8 | 7 | eqcomd 2829 | . . 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 15344 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (∏𝑘 ∈ {𝐶}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
12 | 4 | ancli 551 | . . . . . 6 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ 𝐴)) |
13 | nfv 1915 | . . . . . . . . 9 ⊢ Ⅎ𝑘 𝐶 ∈ 𝐴 | |
14 | 1, 13 | nfan 1900 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ 𝐴) |
15 | nfcsb1v 3909 | . . . . . . . . 9 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 | |
16 | 15 | nfel1 2996 | . . . . . . . 8 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ |
17 | 14, 16 | nfim 1897 | . . . . . . 7 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
18 | eleq1 2902 | . . . . . . . . 9 ⊢ (𝑘 = 𝐶 → (𝑘 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
19 | 18 | anbi2d 630 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐶 ∈ 𝐴))) |
20 | csbeq1a 3899 | . . . . . . . . 9 ⊢ (𝑘 = 𝐶 → 𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
21 | 20 | eleq1d 2899 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → (𝐵 ∈ ℂ ↔ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
22 | 19, 21 | imbi12d 347 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ))) |
23 | 17, 22, 10 | vtoclg1f 3568 | . . . . . 6 ⊢ (𝐶 ∈ 𝐴 → ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
24 | 4, 12, 23 | sylc 65 | . . . . 5 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
25 | prodsns 15328 | . . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝐶}𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
26 | 4, 24, 25 | syl2anc 586 | . . . 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 3915 | . . . 4 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 = 𝐷) |
30 | 26, 29 | eqtrd 2858 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐶}𝐵 = 𝐷) |
31 | 30 | oveq1d 7173 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐶}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵) = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
32 | 11, 31 | eqtrd 2858 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 ⦋csb 3885 ∖ cdif 3935 ∪ cun 3936 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 (class class class)co 7158 Fincfn 8511 ℂcc 10537 · cmul 10544 ∏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: fprodeq0g 15350 fprodsplit1 41881 fprod0 41884 dvmptfprodlem 42236 |
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