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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 4158 | . . . . 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 4809 | . . . . 5 ⊢ (𝜑 → {𝐶} ⊆ 𝐴) |
| 5 | undif 4482 | . . . . 5 ⊢ ({𝐶} ⊆ 𝐴 ↔ ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) | |
| 6 | 4, 5 | sylib 218 | . . . 4 ⊢ (𝜑 → ({𝐶} ∪ (𝐴 ∖ {𝐶})) = 𝐴) |
| 7 | eqidd 2738 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐴) | |
| 8 | 2, 6, 7 | 3eqtrrd 2782 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∖ {𝐶}) ∪ {𝐶})) |
| 9 | 8 | sumeq1d 15736 | . 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 9215 | . . . 4 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝐶}) ∈ Fin) | |
| 14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 ∖ {𝐶}) ∈ Fin) |
| 15 | neldifsnd 4793 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴 ∖ {𝐶})) | |
| 16 | simpl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝐶})) → 𝜑) | |
| 17 | eldifi 4131 | . . . . 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 3929 | . . . . 5 ⊢ (𝜑 → ⦋𝐶 / 𝑘⦌𝐵 = 𝐷) |
| 26 | 25 | eqcomd 2743 | . . . 4 ⊢ (𝜑 → 𝐷 = ⦋𝐶 / 𝑘⦌𝐵) |
| 27 | 3 | ancli 548 | . . . . 5 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ 𝐴)) |
| 28 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑘𝐶 | |
| 29 | nfv 1914 | . . . . . . . 8 ⊢ Ⅎ𝑘 𝐶 ∈ 𝐴 | |
| 30 | 10, 29 | nfan 1899 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐶 ∈ 𝐴) |
| 31 | 28 | nfcsb1 3922 | . . . . . . . 8 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 |
| 32 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑘ℂ | |
| 33 | 31, 32 | nfel 2920 | . . . . . . 7 ⊢ Ⅎ𝑘⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ |
| 34 | 30, 33 | nfim 1896 | . . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝐶 ∈ 𝐴) → ⦋𝐶 / 𝑘⦌𝐵 ∈ ℂ) |
| 35 | eleq1 2829 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → (𝑘 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
| 36 | 35 | anbi2d 630 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝐶 ∈ 𝐴))) |
| 37 | csbeq1a 3913 | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → 𝐵 = ⦋𝐶 / 𝑘⦌𝐵) | |
| 38 | 37 | eleq1d 2826 | . . . . . . 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 2841 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
| 43 | 10, 11, 14, 3, 15, 20, 21, 42 | fsumsplitsn 15780 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐴 ∖ {𝐶}) ∪ {𝐶})𝐵 = (Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 + 𝐷)) |
| 44 | 10, 14, 20 | fsumclf 15774 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 ∈ ℂ) |
| 45 | 44, 42 | addcomd 11463 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵 + 𝐷) = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
| 46 | 9, 43, 45 | 3eqtrd 2781 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (𝐷 + Σ𝑘 ∈ (𝐴 ∖ {𝐶})𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ⦋csb 3899 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 {csn 4626 (class class class)co 7431 Fincfn 8985 ℂcc 11153 + caddc 11158 Σcsu 15722 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 |
| This theorem is referenced by: sticksstones22 42169 unitscyglem4 42199 dvnmul 45958 etransclem35 46284 etransclem44 46293 |
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