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Mirrors > Home > ILE Home > Th. List > sumtp | 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 2395 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
3 | sumtp.3 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
4 | 3 | necomd 2395 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
5 | 2, 4 | nelprd 3558 | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ {𝐴, 𝐵}) |
6 | disjsn 3593 | . . . 4 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ {𝐴, 𝐵}) | |
7 | 5, 6 | sylibr 133 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ∩ {𝐶}) = ∅) |
8 | df-tp 3540 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
9 | 8 | a1i 9 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})) |
10 | sumtp.v | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) | |
11 | 10 | simp1d 994 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
12 | 10 | simp2d 995 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
13 | 10 | simp3d 996 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
14 | sumtp.1 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
15 | 11, 12, 13, 14, 1, 3 | tpfidisj 6824 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ∈ Fin) |
16 | sumtp.c | . . . . 5 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ)) | |
17 | sumtp.e | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) | |
18 | 17 | eleq1d 2209 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐷 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
19 | sumtp.f | . . . . . . . 8 ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) | |
20 | 19 | eleq1d 2209 | . . . . . . 7 ⊢ (𝑘 = 𝐵 → (𝐷 ∈ ℂ ↔ 𝐹 ∈ ℂ)) |
21 | sumtp.g | . . . . . . . 8 ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺) | |
22 | 21 | eleq1d 2209 | . . . . . . 7 ⊢ (𝑘 = 𝐶 → (𝐷 ∈ ℂ ↔ 𝐺 ∈ ℂ)) |
23 | 18, 20, 22 | raltpg 3584 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 ∈ ℂ ↔ (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ))) |
24 | 10, 23 | syl 14 | . . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 ∈ ℂ ↔ (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ))) |
25 | 16, 24 | mpbird 166 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 ∈ ℂ) |
26 | 25 | r19.21bi 2523 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵, 𝐶}) → 𝐷 ∈ ℂ) |
27 | 7, 9, 15, 26 | fsumsplit 11208 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = (Σ𝑘 ∈ {𝐴, 𝐵}𝐷 + Σ𝑘 ∈ {𝐶}𝐷)) |
28 | 3simpa 979 | . . . . 5 ⊢ ((𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ) → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) | |
29 | 16, 28 | syl 14 | . . . 4 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ)) |
30 | 3simpa 979 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) | |
31 | 10, 30 | syl 14 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
32 | 17, 19, 29, 31, 14 | sumpr 11214 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 + 𝐹)) |
33 | 16 | simp3d 996 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ ℂ) |
34 | 21 | sumsn 11212 | . . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐺 ∈ ℂ) → Σ𝑘 ∈ {𝐶}𝐷 = 𝐺) |
35 | 13, 33, 34 | syl2anc 409 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐶}𝐷 = 𝐺) |
36 | 32, 35 | oveq12d 5800 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ {𝐴, 𝐵}𝐷 + Σ𝑘 ∈ {𝐶}𝐷) = ((𝐸 + 𝐹) + 𝐺)) |
37 | 27, 36 | eqtrd 2173 | 1 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 ∀wral 2417 ∪ cun 3074 ∩ cin 3075 ∅c0 3368 {csn 3532 {cpr 3533 {ctp 3534 (class class class)co 5782 ℂcc 7642 + caddc 7647 Σcsu 11154 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-tp 3540 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-en 6643 df-dom 6644 df-fin 6645 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-ihash 10554 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155 |
This theorem is referenced by: (None) |
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