Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sumpr | Structured version Visualization version GIF version |
Description: A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.) |
Ref | Expression |
---|---|
sumpr.1 | ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) |
sumpr.2 | ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸) |
sumpr.3 | ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ)) |
sumpr.4 | ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
sumpr.5 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
sumpr | ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumpr.5 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | disjsn2 4650 | . . . 4 ⊢ (𝐴 ≠ 𝐵 → ({𝐴} ∩ {𝐵}) = ∅) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → ({𝐴} ∩ {𝐵}) = ∅) |
4 | df-pr 4572 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})) |
6 | prfi 8795 | . . . 4 ⊢ {𝐴, 𝐵} ∈ Fin | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ Fin) |
8 | sumpr.3 | . . . . 5 ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ)) | |
9 | sumpr.4 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) | |
10 | sumpr.1 | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) | |
11 | 10 | eleq1d 2899 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → (𝐶 ∈ ℂ ↔ 𝐷 ∈ ℂ)) |
12 | sumpr.2 | . . . . . . . 8 ⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸) | |
13 | 12 | eleq1d 2899 | . . . . . . 7 ⊢ (𝑘 = 𝐵 → (𝐶 ∈ ℂ ↔ 𝐸 ∈ ℂ)) |
14 | 11, 13 | ralprg 4634 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑘 ∈ {𝐴, 𝐵}𝐶 ∈ ℂ ↔ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))) |
15 | 9, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → (∀𝑘 ∈ {𝐴, 𝐵}𝐶 ∈ ℂ ↔ (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))) |
16 | 8, 15 | mpbird 259 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ {𝐴, 𝐵}𝐶 ∈ ℂ) |
17 | 16 | r19.21bi 3210 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ ℂ) |
18 | 3, 5, 7, 17 | fsumsplit 15099 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (Σ𝑘 ∈ {𝐴}𝐶 + Σ𝑘 ∈ {𝐵}𝐶)) |
19 | 9 | simpld 497 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
20 | 8 | simpld 497 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
21 | 10 | sumsn 15103 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐷 ∈ ℂ) → Σ𝑘 ∈ {𝐴}𝐶 = 𝐷) |
22 | 19, 20, 21 | syl2anc 586 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴}𝐶 = 𝐷) |
23 | 9 | simprd 498 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
24 | 8 | simprd 498 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
25 | 12 | sumsn 15103 | . . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐸 ∈ ℂ) → Σ𝑘 ∈ {𝐵}𝐶 = 𝐸) |
26 | 23, 24, 25 | syl2anc 586 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ {𝐵}𝐶 = 𝐸) |
27 | 22, 26 | oveq12d 7176 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ {𝐴}𝐶 + Σ𝑘 ∈ {𝐵}𝐶) = (𝐷 + 𝐸)) |
28 | 18, 27 | eqtrd 2858 | 1 ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 {cpr 4571 (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: sumtp 15106 ehl2eudis 24027 sge0pr 42683 nnsum3primes4 43960 nnsum3primesgbe 43964 |
Copyright terms: Public domain | W3C validator |