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Mirrors > Home > MPE Home > Th. List > fsumsplit1 | Structured version Visualization version GIF version |
Description: Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fsumsplit1.kph | ⊢ Ⅎ𝑘𝜑 |
fsumsplit1.kd | ⊢ Ⅎ𝑘𝐷 |
fsumsplit1.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumsplit1.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
fsumsplit1.c | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
fsumsplit1.bd | ⊢ (𝑘 = 𝐶 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
fsumsplit1 | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 4114 | . . . . 5 ⊢ ((𝐴 ∖ {𝐶}) ∪ {𝐶}) = ({𝐶} ∪ (𝐴 ∖ {𝐶})) | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → ((𝐴 ∖ {𝐶}) ∪ {𝐶}) = ({𝐶} ∪ (𝐴 ∖ {𝐶}))) |
3 | fsumsplit1.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
4 | 3 | snssd 4770 | . . . . 5 ⊢ (𝜑 → {𝐶} ⊆ 𝐴) |
5 | undif 4442 | . . . . 5 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) | |
6 | 4, 5 | sylib 217 | . . . 4 ⊢ (𝜑 → ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) |
7 | eqidd 2734 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐴) | |
8 | 2, 6, 7 | 3eqtrrd 2778 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∖ {𝐶}) ∪ {𝐶})) |
9 | 8 | sumeq1d 15591 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ((𝐴 ∖ {𝐶}) ∪ {𝐶})𝐵) |
10 | fsumsplit1.kph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
11 | fsumsplit1.kd | . . 3 ⊢ Ⅎ𝑘𝐷 | |
12 | fsumsplit1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
13 | diffi 9126 | . . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝐶}) ∈ Fin) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∖ {𝐶}) ∈ Fin) |
15 | neldifsnd 4754 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴 ∖ {𝐶})) | |
16 | simpl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝜑) | |
17 | eldifi 4087 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∖ {𝐶}) → 𝑘 ∈ 𝐴) | |
18 | 17 | adantl 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝑘 ∈ 𝐴) |
19 | fsumsplit1.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
20 | 16, 18, 19 | syl2anc 585 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝐵 ∈ ℂ) |
21 | fsumsplit1.bd | . . 3 ⊢ (𝑘 = 𝐶 → 𝐵 = 𝐷) | |
22 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑘𝐷) |
23 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝑘 = 𝐶) | |
24 | 23, 21 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷) |
25 | 10, 22, 3, 24 | csbiedf 3887 | . . . . 5 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 = 𝐷) |
26 | 25 | eqcomd 2739 | . . . 4 ⊢ (𝜑 → 𝐷 = ⦋𝐶 / 𝑘⦌𝐵) |
27 | 3 | ancli 550 | . . . . 5 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ 𝐴)) |
28 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑘𝐶 | |
29 | nfv 1918 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝐶 ∈ 𝐴 | |
30 | 10, 29 | nfan 1903 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ 𝐴) |
31 | 28 | nfcsb1 3880 | . . . . . . . 8 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 |
32 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑘ℂ | |
33 | 31, 32 | nfel 2918 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ |
34 | 30, 33 | nfim 1900 | . . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
35 | eleq1 2822 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → (𝑘 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
36 | 35 | anbi2d 630 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐶 ∈ 𝐴))) |
37 | csbeq1a 3870 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → 𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
38 | 37 | eleq1d 2819 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → (𝐵 ∈ ℂ ↔ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
39 | 36, 38 | imbi12d 345 | . . . . . 6 ⊢ (𝑘 = 𝐶 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ))) |
40 | 28, 34, 39, 19 | vtoclgf 3522 | . . . . 5 ⊢ (𝐶 ∈ 𝐴 → ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
41 | 3, 27, 40 | sylc 65 | . . . 4 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
42 | 26, 41 | eqeltrd 2834 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
43 | 10, 11, 14, 3, 15, 20, 21, 42 | fsumsplitsn 15634 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐴 ∖ {𝐶}) ∪ {𝐶})𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 + 𝐷)) |
44 | 10, 14, 20 | fsumclf 15628 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 ∈ ℂ) |
45 | 44, 42 | addcomd 11362 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 + 𝐷) = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
46 | 9, 43, 45 | 3eqtrd 2777 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 ⦋csb 3856 ∖ cdif 3908 ∪ cun 3909 ⊆ wss 3911 {csn 4587 (class class class)co 7358 Fincfn 8886 ℂcc 11054 + caddc 11059 Σcsu 15576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-sum 15577 |
This theorem is referenced by: sticksstones22 40622 dvnmul 44270 etransclem35 44596 etransclem44 44605 |
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