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Mirrors > Home > MPE Home > Th. List > Mathboxes > sumpair | Structured version Visualization version GIF version |
Description: Sum of two distinct complex values. The class expression for 𝐴 and 𝐵 normally contain free variable 𝑘 to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
sumpair.1 | ⊢ (𝜑 → Ⅎ𝑘𝐷) |
sumpair.3 | ⊢ (𝜑 → Ⅎ𝑘𝐸) |
sumupair.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sumupair.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
sumupair.3 | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
sumupair.4 | ⊢ (𝜑 → 𝐸 ∈ ℂ) |
sumupair.5 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
sumupair.8 | ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) |
sumupair.9 | ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
sumpair | ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumupair.5 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | disjsn2 4614 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
4 | df-pr 4530 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})) |
6 | prfi 8924 | . . . 4 ⊢ {𝐴, 𝐵} ∈ Fin | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin) |
8 | elpri 4549 | . . . 4 ⊢ (𝑘 ∈ {𝐴, 𝐵} → (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) | |
9 | sumupair.8 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) | |
10 | sumupair.3 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
11 | 10 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 ∈ ℂ) |
12 | 9, 11 | eqeltrd 2831 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 ∈ ℂ) |
13 | sumupair.9 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) | |
14 | sumupair.4 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℂ) | |
15 | 14 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐸 ∈ ℂ) |
16 | 13, 15 | eqeltrd 2831 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 ∈ ℂ) |
17 | 12, 16 | jaodan 958 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 = 𝐴 ∨ 𝑘 = 𝐵)) → 𝐶 ∈ ℂ) |
18 | 8, 17 | sylan2 596 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ ℂ) |
19 | 3, 5, 7, 18 | fsumsplit 15269 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (Σ𝑘 ∈ {𝐴}𝐶 + Σ𝑘 ∈ {𝐵}𝐶)) |
20 | sumpair.1 | . . . 4 ⊢ (𝜑 → Ⅎ𝑘𝐷) | |
21 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
22 | sumupair.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
23 | 20, 21, 9, 22, 10 | sumsnd 42183 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴}𝐶 = 𝐷) |
24 | sumpair.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑘𝐸) | |
25 | sumupair.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
26 | 24, 21, 13, 25, 14 | sumsnd 42183 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐵}𝐶 = 𝐸) |
27 | 23, 26 | oveq12d 7209 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ {𝐴}𝐶 + Σ𝑘 ∈ {𝐵}𝐶) = (𝐷 + 𝐸)) |
28 | 19, 27 | eqtrd 2771 | 1 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 Ⅎwnfc 2877 ≠ wne 2932 ∪ cun 3851 ∩ cin 3852 ∅c0 4223 {csn 4527 {cpr 4529 (class class class)co 7191 Fincfn 8604 ℂcc 10692 + caddc 10697 Σcsu 15214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-fz 13061 df-fzo 13204 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-clim 15014 df-sum 15215 |
This theorem is referenced by: refsum2cnlem1 42194 |
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