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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 3073 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
3 | sumtp.3 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
4 | 3 | necomd 3073 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
5 | 2, 4 | nelprd 4598 | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ {𝐴, 𝐵}) |
6 | disjsn 4649 | . . . 4 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ {𝐴, 𝐵}) | |
7 | 5, 6 | sylibr 236 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
8 | df-tp 4574 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})) |
10 | tpfi 8796 | . . . 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 2899 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐷 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
16 | sumtp.f | . . . . . . . 8 ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) | |
17 | 16 | eleq1d 2899 | . . . . . . 7 ⊢ (𝑘 = 𝐵 → (𝐷 ∈ ℂ ↔ 𝐹 ∈ ℂ)) |
18 | sumtp.g | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺) | |
19 | 18 | eleq1d 2899 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → (𝐷 ∈ ℂ ↔ 𝐺 ∈ ℂ)) |
20 | 15, 17, 19 | raltpg 4636 | . . . . . 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 259 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 ∈ ℂ) |
23 | 22 | r19.21bi 3210 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵, 𝐶}) → 𝐷 ∈ ℂ) |
24 | 7, 9, 11, 23 | fsumsplit 15099 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = (Σ𝑘 ∈ {𝐴, 𝐵}𝐷 + Σ𝑘 ∈ {𝐶}𝐷)) |
25 | 3simpa 1144 | . . . . 5 ⊢ ((𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ) → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) | |
26 | 12, 25 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) |
27 | 3simpa 1144 | . . . . 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 15105 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 + 𝐹)) |
31 | 13 | simp3d 1140 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
32 | 12 | simp3d 1140 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ ℂ) |
33 | 18 | sumsn 15103 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐺 ∈ ℂ) → Σ𝑘 ∈ {𝐶}𝐷 = 𝐺) |
34 | 31, 32, 33 | syl2anc 586 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐶}𝐷 = 𝐺) |
35 | 30, 34 | oveq12d 7176 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ {𝐴, 𝐵}𝐷 + Σ𝑘 ∈ {𝐶}𝐷) = ((𝐸 + 𝐹) + 𝐺)) |
36 | 24, 35 | eqtrd 2858 | 1 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 {cpr 4571 {ctp 4573 (class class class)co 7158 Fincfn 8511 ℂcc 10537 + caddc 10542 Σcsu 15044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 |
This theorem is referenced by: hgt750lemb 31929 tgoldbachgt 31936 nnsum4primesodd 43968 nnsum4primesoddALTV 43969 |
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