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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 4168 | . . . . 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 4814 | . . . . 5 ⊢ (𝜑 → {𝐶} ⊆ 𝐴) |
5 | undif 4488 | . . . . 5 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) | |
6 | 4, 5 | sylib 218 | . . . 4 ⊢ (𝜑 → ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) |
7 | eqidd 2736 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐴) | |
8 | 2, 6, 7 | 3eqtrrd 2780 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∖ {𝐶}) ∪ {𝐶})) |
9 | 8 | sumeq1d 15733 | . 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 9214 | . . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝐶}) ∈ Fin) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∖ {𝐶}) ∈ Fin) |
15 | neldifsnd 4798 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴 ∖ {𝐶})) | |
16 | simpl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝜑) | |
17 | eldifi 4141 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∖ {𝐶}) → 𝑘 ∈ 𝐴) | |
18 | 17 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝑘 ∈ 𝐴) |
19 | fsumsplit1.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
20 | 16, 18, 19 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝐵 ∈ ℂ) |
21 | fsumsplit1.bd | . . 3 ⊢ (𝑘 = 𝐶 → 𝐵 = 𝐷) | |
22 | 11 | a1i 11 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑘𝐷) |
23 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝑘 = 𝐶) | |
24 | 23, 21 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷) |
25 | 10, 22, 3, 24 | csbiedf 3939 | . . . . 5 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 = 𝐷) |
26 | 25 | eqcomd 2741 | . . . 4 ⊢ (𝜑 → 𝐷 = ⦋𝐶 / 𝑘⦌𝐵) |
27 | 3 | ancli 548 | . . . . 5 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ 𝐴)) |
28 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑘𝐶 | |
29 | nfv 1912 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝐶 ∈ 𝐴 | |
30 | 10, 29 | nfan 1897 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ 𝐴) |
31 | 28 | nfcsb1 3932 | . . . . . . . 8 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 |
32 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑘ℂ | |
33 | 31, 32 | nfel 2918 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ |
34 | 30, 33 | nfim 1894 | . . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
35 | eleq1 2827 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → (𝑘 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
36 | 35 | anbi2d 630 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐶 ∈ 𝐴))) |
37 | csbeq1a 3922 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → 𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
38 | 37 | eleq1d 2824 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → (𝐵 ∈ ℂ ↔ ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ)) |
39 | 36, 38 | imbi12d 344 | . . . . . 6 ⊢ (𝑘 = 𝐶 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ))) |
40 | 28, 34, 39, 19 | vtoclgf 3569 | . . . . 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 2839 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
43 | 10, 11, 14, 3, 15, 20, 21, 42 | fsumsplitsn 15777 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐴 ∖ {𝐶}) ∪ {𝐶})𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 + 𝐷)) |
44 | 10, 14, 20 | fsumclf 15771 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 ∈ ℂ) |
45 | 44, 42 | addcomd 11461 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 + 𝐷) = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
46 | 9, 43, 45 | 3eqtrd 2779 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (𝐷 + Σ𝑘 ∈ (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 ⊆ wss 3963 {csn 4631 (class class class)co 7431 Fincfn 8984 ℂcc 11151 + caddc 11156 Σcsu 15719 |
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-sum 15720 |
This theorem is referenced by: sticksstones22 42150 unitscyglem4 42180 dvnmul 45899 etransclem35 46225 etransclem44 46234 |
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