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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 2996 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
| 3 | sumtp.3 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
| 4 | 3 | necomd 2996 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
| 5 | 2, 4 | nelprd 4657 | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ {𝐴, 𝐵}) |
| 6 | disjsn 4711 | . . . 4 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ {𝐴, 𝐵}) | |
| 7 | 5, 6 | sylibr 234 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
| 8 | df-tp 4631 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
| 9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})) |
| 10 | tpfi 9365 | . . . 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 2826 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐷 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
| 16 | sumtp.f | . . . . . . . 8 ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) | |
| 17 | 16 | eleq1d 2826 | . . . . . . 7 ⊢ (𝑘 = 𝐵 → (𝐷 ∈ ℂ ↔ 𝐹 ∈ ℂ)) |
| 18 | sumtp.g | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺) | |
| 19 | 18 | eleq1d 2826 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → (𝐷 ∈ ℂ ↔ 𝐺 ∈ ℂ)) |
| 20 | 15, 17, 19 | raltpg 4698 | . . . . . 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 257 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 ∈ ℂ) |
| 23 | 22 | r19.21bi 3251 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵, 𝐶}) → 𝐷 ∈ ℂ) |
| 24 | 7, 9, 11, 23 | fsumsplit 15777 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = (Σ𝑘 ∈ {𝐴, 𝐵}𝐷 + Σ𝑘 ∈ {𝐶}𝐷)) |
| 25 | 3simpa 1149 | . . . . 5 ⊢ ((𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ) → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) | |
| 26 | 12, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) |
| 27 | 3simpa 1149 | . . . . 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 15784 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 + 𝐹)) |
| 31 | 13 | simp3d 1145 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
| 32 | 12 | simp3d 1145 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ ℂ) |
| 33 | 18 | sumsn 15782 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐺 ∈ ℂ) → Σ𝑘 ∈ {𝐶}𝐷 = 𝐺) |
| 34 | 31, 32, 33 | syl2anc 584 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐶}𝐷 = 𝐺) |
| 35 | 30, 34 | oveq12d 7449 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ {𝐴, 𝐵}𝐷 + Σ𝑘 ∈ {𝐶}𝐷) = ((𝐸 + 𝐹) + 𝐺)) |
| 36 | 24, 35 | eqtrd 2777 | 1 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 {csn 4626 {cpr 4628 {ctp 4630 (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-tp 4631 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-2o 8507 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: hgt750lemb 34671 tgoldbachgt 34678 nnsum4primesodd 47783 nnsum4primesoddALTV 47784 |
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