sumtp - Intuitionistic Logic Explorer

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Theorem sumtp 11424

Description: A sum over a triple is the sum of the elements. (Contributed by AV, 24-Jul-2020.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
sumtp	⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝐶,𝑘 𝑘,𝐸 𝑘,𝐹 𝑘,𝐺 𝜑,𝑘 𝑘,𝑉 𝑘,𝑊 𝑘,𝑋 Allowed substitution hint: 𝐷(𝑘)

**Proof of Theorem sumtp**
Step	Hyp	Ref	Expression
1		sumtp.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
2	1	necomd 2433	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
3		sumtp.3	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
4	3	necomd 2433	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
5	2, 4	nelprd 3620	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ {𝐴, 𝐵})
6		disjsn 3656	. . . 4 ⊢ (({𝐴, 𝐵} ∩ {𝐶}) = ∅ ↔ ¬ 𝐶 ∈ {𝐴, 𝐵})
7	5, 6	sylibr 134	. . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ∩ {𝐶}) = ∅)
8		df-tp 3602	. . . 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 1009	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12	10	simp2d 1010	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
13	10	simp3d 1011	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
14		sumtp.1	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
15	11, 12, 13, 14, 1, 3	tpfidisj 6929	. . 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 2246	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐷 ∈ ℂ ↔ 𝐸 ∈ ℂ))
19		sumtp.f	. . . . . . . 8 ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹)
20	19	eleq1d 2246	. . . . . . 7 ⊢ (𝑘 = 𝐵 → (𝐷 ∈ ℂ ↔ 𝐹 ∈ ℂ))
21		sumtp.g	. . . . . . . 8 ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺)
22	21	eleq1d 2246	. . . . . . 7 ⊢ (𝑘 = 𝐶 → (𝐷 ∈ ℂ ↔ 𝐺 ∈ ℂ))
23	18, 20, 22	raltpg 3647	. . . . . 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 167	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 ∈ ℂ)
26	25	r19.21bi 2565	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵, 𝐶}) → 𝐷 ∈ ℂ)
27	7, 9, 15, 26	fsumsplit 11417	. 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = (Σ𝑘 ∈ {𝐴, 𝐵}𝐷 + Σ𝑘 ∈ {𝐶}𝐷))
28		3simpa 994	. . . . 5 ⊢ ((𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ ∧ 𝐺 ∈ ℂ) → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ))
29	16, 28	syl 14	. . . 4 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ))
30		3simpa 994	. . . . 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 11423	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 + 𝐹))
33	16	simp3d 1011	. . . 4 ⊢ (𝜑 → 𝐺 ∈ ℂ)
34	21	sumsn 11421	. . . 4 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐺 ∈ ℂ) → Σ𝑘 ∈ {𝐶}𝐷 = 𝐺)
35	13, 33, 34	syl2anc 411	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐶}𝐷 = 𝐺)
36	32, 35	oveq12d 5895	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ {𝐴, 𝐵}𝐷 + Σ𝑘 ∈ {𝐶}𝐷) = ((𝐸 + 𝐹) + 𝐺))
37	27, 36	eqtrd 2210	1 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 + 𝐹) + 𝐺))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 ∪ cun 3129 ∩ cin 3130 ∅c0 3424 {csn 3594 {cpr 3595 {ctp 3596 (class class class)co 5877 ℂcc 7811 + caddc 7816 Σcsu 11363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-tp 3602 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-en 6743 df-dom 6744 df-fin 6745 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-ihash 10758 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-sumdc 11364

This theorem is referenced by: (None)

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