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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 39825 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 4040 | . . . 4 ⊢ ({𝐶} ∩ (𝐴 ∖ {𝐶})) = ∅ | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → ({𝐶} ∩ (𝐴 ∖ {𝐶})) = ∅) |
4 | fprodsplit1f.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
5 | snssi 4339 | . . . . . 6 ⊢ (𝐶 ∈ 𝐴 → {𝐶} ⊆ 𝐴) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → {𝐶} ⊆ 𝐴) |
7 | undif 4049 | . . . . 5 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) | |
8 | 6, 7 | sylib 208 | . . . 4 ⊢ (𝜑 → ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) |
9 | 8 | eqcomd 2628 | . . 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 14719 | . 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 3554 | . . . . . 6 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 = 𝐷) |
16 | 15 | eqcomd 2628 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = ⦋𝐶 / 𝑘⦌𝐵) |
17 | 4 | ancli 574 | . . . . . . . 8 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ 𝐴)) |
18 | nfcv 2764 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐶 | |
19 | nfv 1843 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝐶 ∈ 𝐴 | |
20 | 1, 19 | nfan 1828 | . . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ 𝐴) |
21 | 18 | nfcsb1 3548 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 |
22 | nfcv 2764 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘ℂ | |
23 | 21, 22 | nfel 2777 | . . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ |
24 | 20, 23 | nfim 1825 | . . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
25 | eleq1 2689 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝐶 → (𝑘 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
26 | 25 | anbi2d 740 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐶 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐶 ∈ 𝐴))) |
27 | csbeq1a 3542 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝐶 → 𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
28 | 27 | eleq1d 2686 | . . . . . . . . . 10 ⊢ (𝑘 = 𝐶 → (𝐵 ∈ ℂ ↔ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
29 | 26, 28 | imbi12d 334 | . . . . . . . . 9 ⊢ (𝑘 = 𝐶 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ))) |
30 | 18, 24, 29, 11 | vtoclgf 3264 | . . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
31 | 4, 17, 30 | sylc 65 | . . . . . . 7 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
32 | 16, 31 | eqeltrd 2701 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
33 | 15, 32 | eqeltrd 2701 | . . . . 5 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
34 | prodsns 14702 | . . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝐶}𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
35 | 4, 33, 34 | syl2anc 693 | . . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {𝐶}𝐵 = ⦋𝐶 / 𝑘⦌𝐵) |
36 | 35, 15 | eqtrd 2656 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ {𝐶}𝐵 = 𝐷) |
37 | 36 | oveq1d 6665 | . 2 ⊢ (𝜑 → (∏𝑘 ∈ {𝐶}𝐵 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵) = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
38 | 12, 37 | eqtrd 2656 | 1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ⦋csb 3533 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 (class class class)co 6650 Fincfn 7955 ℂcc 9934 · cmul 9941 ∏cprod 14635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-prod 14636 |
This theorem is referenced by: fprodeq0g 14725 fprodsplit1 39825 fprod0 39828 dvmptfprodlem 40159 |
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