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Mirrors > Home > ILE Home > Th. List > sumpr | GIF version |
Description: A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.) |
Ref | Expression |
---|---|
sumpr.1 | ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) |
sumpr.2 | ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸) |
sumpr.3 | ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ)) |
sumpr.4 | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
sumpr.5 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
sumpr | ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumpr.5 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | disjsn2 3581 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
4 | df-pr 3529 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
5 | 4 | a1i 9 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})) |
6 | sumpr.4 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) | |
7 | 6 | simpld 111 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
8 | 6 | simprd 113 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
9 | prfidisj 6808 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ∈ Fin) | |
10 | 7, 8, 1, 9 | syl3anc 1216 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin) |
11 | sumpr.3 | . . . . 5 ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ)) | |
12 | sumpr.1 | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) | |
13 | 12 | eleq1d 2206 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐶 ∈ ℂ ↔ 𝐷 ∈ ℂ)) |
14 | sumpr.2 | . . . . . . . 8 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸) | |
15 | 14 | eleq1d 2206 | . . . . . . 7 ⊢ (𝑘 = 𝐵 → (𝐶 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
16 | 13, 15 | ralprg 3569 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑘 ∈ {𝐴, 𝐵}𝐶 ∈ ℂ ↔ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))) |
17 | 6, 16 | syl 14 | . . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ {𝐴, 𝐵}𝐶 ∈ ℂ ↔ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))) |
18 | 11, 17 | mpbird 166 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ {𝐴, 𝐵}𝐶 ∈ ℂ) |
19 | 18 | r19.21bi 2518 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ ℂ) |
20 | 3, 5, 10, 19 | fsumsplit 11169 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (Σ𝑘 ∈ {𝐴}𝐶 + Σ𝑘 ∈ {𝐵}𝐶)) |
21 | 11 | simpld 111 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
22 | 12 | sumsn 11173 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → Σ𝑘 ∈ {𝐴}𝐶 = 𝐷) |
23 | 7, 21, 22 | syl2anc 408 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴}𝐶 = 𝐷) |
24 | 11 | simprd 113 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
25 | 14 | sumsn 11173 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐸 ∈ ℂ) → Σ𝑘 ∈ {𝐵}𝐶 = 𝐸) |
26 | 8, 24, 25 | syl2anc 408 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐵}𝐶 = 𝐸) |
27 | 23, 26 | oveq12d 5785 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ {𝐴}𝐶 + Σ𝑘 ∈ {𝐵}𝐶) = (𝐷 + 𝐸)) |
28 | 20, 27 | eqtrd 2170 | 1 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 ∀wral 2414 ∪ cun 3064 ∩ cin 3065 ∅c0 3358 {csn 3522 {cpr 3523 (class class class)co 5767 Fincfn 6627 ℂcc 7611 + caddc 7616 Σcsu 11115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-frec 6281 df-1o 6306 df-oadd 6310 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-fz 9784 df-fzo 9913 df-seqfrec 10212 df-exp 10286 df-ihash 10515 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-clim 11041 df-sumdc 11116 |
This theorem is referenced by: sumtp 11176 |
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