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Mirrors > Home > MPE Home > Th. List > sumtp | Structured version Visualization version GIF version |
Description: A sum over a triple is the sum of the elements. (Contributed by AV, 24-Jul-2020.) |
Ref | Expression |
---|---|
sumtp.e | ⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) |
sumtp.f | ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) |
sumtp.g | ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺) |
sumtp.c | ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ)) |
sumtp.v | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) |
sumtp.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
sumtp.2 | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
sumtp.3 | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Ref | Expression |
---|---|
sumtp | ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumtp.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
2 | 1 | necomd 2986 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
3 | sumtp.3 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
4 | 3 | necomd 2986 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
5 | 2, 4 | nelprd 4654 | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ {𝐴, 𝐵}) |
6 | disjsn 4710 | . . . 4 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ {𝐴, 𝐵}) | |
7 | 5, 6 | sylibr 233 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
8 | df-tp 4628 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})) |
10 | tpfi 9360 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} ∈ Fin | |
11 | 10 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin) |
12 | sumtp.c | . . . . 5 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ)) | |
13 | sumtp.v | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) | |
14 | sumtp.e | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) | |
15 | 14 | eleq1d 2811 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐷 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
16 | sumtp.f | . . . . . . . 8 ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) | |
17 | 16 | eleq1d 2811 | . . . . . . 7 ⊢ (𝑘 = 𝐵 → (𝐷 ∈ ℂ ↔ 𝐹 ∈ ℂ)) |
18 | sumtp.g | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺) | |
19 | 18 | eleq1d 2811 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → (𝐷 ∈ ℂ ↔ 𝐺 ∈ ℂ)) |
20 | 15, 17, 19 | raltpg 4697 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 ∈ ℂ ↔ (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ))) |
21 | 13, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 ∈ ℂ ↔ (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ))) |
22 | 12, 21 | mpbird 256 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 ∈ ℂ) |
23 | 22 | r19.21bi 3239 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵, 𝐶}) → 𝐷 ∈ ℂ) |
24 | 7, 9, 11, 23 | fsumsplit 15740 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = (Σ𝑘 ∈ {𝐴, 𝐵}𝐷 + Σ𝑘 ∈ {𝐶}𝐷)) |
25 | 3simpa 1145 | . . . . 5 ⊢ ((𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ) → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) | |
26 | 12, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) |
27 | 3simpa 1145 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) | |
28 | 13, 27 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
29 | sumtp.1 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
30 | 14, 16, 26, 28, 29 | sumpr 15747 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 + 𝐹)) |
31 | 13 | simp3d 1141 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
32 | 12 | simp3d 1141 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ ℂ) |
33 | 18 | sumsn 15745 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐺 ∈ ℂ) → Σ𝑘 ∈ {𝐶}𝐷 = 𝐺) |
34 | 31, 32, 33 | syl2anc 582 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐶}𝐷 = 𝐺) |
35 | 30, 34 | oveq12d 7434 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ {𝐴, 𝐵}𝐷 + Σ𝑘 ∈ {𝐶}𝐷) = ((𝐸 + 𝐹) + 𝐺)) |
36 | 24, 35 | eqtrd 2766 | 1 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∪ cun 3944 ∩ cin 3945 ∅c0 4322 {csn 4623 {cpr 4625 {ctp 4627 (class class class)co 7416 Fincfn 8966 ℂcc 11147 + caddc 11152 Σcsu 15685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-z 12605 df-uz 12869 df-rp 13023 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-sum 15686 |
This theorem is referenced by: hgt750lemb 34515 tgoldbachgt 34522 nnsum4primesodd 47404 nnsum4primesoddALTV 47405 |
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