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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 38660 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 3992 | . . . 4 ⊢ ({𝐶} ∩ (𝐴 ∖ {𝐶})) = ∅ | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐶} ∩ (𝐴 ∖ {𝐶})) = ∅) |
4 | fprodsplit1f.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
5 | snssi 4280 | . . . . . 6 ⊢ (𝐶 ∈ 𝐴 → {𝐶} ⊆ 𝐴) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → {𝐶} ⊆ 𝐴) |
7 | undif 4001 | . . . . 5 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) | |
8 | 6, 7 | sylib 207 | . . . 4 ⊢ (𝜑 → ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) |
9 | 8 | eqcomd 2616 | . . 3 ⊢ (𝜑 → 𝐴 = ({𝐶} ∪ (𝐴 ∖ {𝐶}))) |
10 | fprodsplit1f.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
11 | fprodsplit1f.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
12 | 1, 3, 9, 10, 11 | fprodsplitf 14558 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (∏𝑘 ∈ {𝐶}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
13 | fprodsplit1f.fk | . . . . . . 7 ⊢ (𝜑 → Ⅎ𝑘𝐷) | |
14 | fprodsplit1f.d | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷) | |
15 | 1, 13, 4, 14 | csbiedf 3520 | . . . . . 6 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 = 𝐷) |
16 | 15 | eqcomd 2616 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = ⦋𝐶 / 𝑘⦌𝐵) |
17 | 4 | ancli 572 | . . . . . . . 8 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ 𝐴)) |
18 | nfcv 2751 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐶 | |
19 | nfv 1830 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝐶 ∈ 𝐴 | |
20 | 1, 19 | nfan 1816 | . . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ 𝐴) |
21 | 18 | nfcsb1 3514 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 |
22 | nfcv 2751 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘ℂ | |
23 | 21, 22 | nfel 2763 | . . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ |
24 | 20, 23 | nfim 1813 | . . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
25 | eleq1 2676 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝐶 → (𝑘 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
26 | 25 | anbi2d 736 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐶 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐶 ∈ 𝐴))) |
27 | csbeq1a 3508 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝐶 → 𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
28 | 27 | eleq1d 2672 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐶 → (𝐵 ∈ ℂ ↔ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
29 | 26, 28 | imbi12d 333 | . . . . . . . . 9 ⊢ (𝑘 = 𝐶 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ))) |
30 | 18, 24, 29, 11 | vtoclgf 3237 | . . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
31 | 4, 17, 30 | sylc 63 | . . . . . . 7 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
32 | 16, 31 | eqeltrd 2688 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
33 | 15, 32 | eqeltrd 2688 | . . . . 5 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
34 | prodsns 14541 | . . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝐶}𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
35 | 4, 33, 34 | syl2anc 691 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {𝐶}𝐵 = ⦋𝐶 / 𝑘⦌𝐵) |
36 | 35, 15 | eqtrd 2644 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐶}𝐵 = 𝐷) |
37 | 36 | oveq1d 6564 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐶}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵) = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
38 | 12, 37 | eqtrd 2644 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 ⦋csb 3499 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 (class class class)co 6549 Fincfn 7841 ℂcc 9813 · cmul 9820 ∏cprod 14474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-prod 14475 |
This theorem is referenced by: fprodeq0g 14564 fprodsplit1 38660 fprod0 38663 dvmptfprodlem 38834 |
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