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Mirrors > Home > ILE Home > Th. List > addcnsr | GIF version |
Description: Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) |
Ref | Expression |
---|---|
addcnsr | ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addclsr 6930 | . . . 4 ⊢ ((𝐴 ∈ R ∧ 𝐶 ∈ R) → (𝐴 +R 𝐶) ∈ R) | |
2 | 1 | ad2ant2r 492 | . . 3 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (𝐴 +R 𝐶) ∈ R) |
3 | addclsr 6930 | . . . 4 ⊢ ((𝐵 ∈ R ∧ 𝐷 ∈ R) → (𝐵 +R 𝐷) ∈ R) | |
4 | 3 | ad2ant2l 491 | . . 3 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (𝐵 +R 𝐷) ∈ R) |
5 | opelxpi 4394 | . . 3 ⊢ (((𝐴 +R 𝐶) ∈ R ∧ (𝐵 +R 𝐷) ∈ R) → 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉 ∈ (R × R)) | |
6 | 2, 4, 5 | syl2anc 403 | . 2 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉 ∈ (R × R)) |
7 | simpll 495 | . . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑤 = 𝐴) | |
8 | simprl 497 | . . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑢 = 𝐶) | |
9 | 7, 8 | oveq12d 5550 | . . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑤 +R 𝑢) = (𝐴 +R 𝐶)) |
10 | simplr 496 | . . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑣 = 𝐵) | |
11 | simprr 498 | . . . 4 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑓 = 𝐷) | |
12 | 10, 11 | oveq12d 5550 | . . 3 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → (𝑣 +R 𝑓) = (𝐵 +R 𝐷)) |
13 | 9, 12 | opeq12d 3578 | . 2 ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉 = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉) |
14 | df-add 6992 | . . 3 ⊢ + = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} | |
15 | df-c 6987 | . . . . . . 7 ⊢ ℂ = (R × R) | |
16 | 15 | eleq2i 2145 | . . . . . 6 ⊢ (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R)) |
17 | 15 | eleq2i 2145 | . . . . . 6 ⊢ (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R)) |
18 | 16, 17 | anbi12i 447 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R))) |
19 | 18 | anbi1i 445 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉)) ↔ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))) |
20 | 19 | oprabbii 5580 | . . 3 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} |
21 | 14, 20 | eqtri 2101 | . 2 ⊢ + = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} |
22 | 6, 13, 21 | ovi3 5657 | 1 ⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∃wex 1421 ∈ wcel 1433 〈cop 3401 × cxp 4361 (class class class)co 5532 {coprab 5533 Rcnr 6487 +R cplr 6491 ℂcc 6979 + caddc 6984 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-iplp 6658 df-enr 6903 df-nr 6904 df-plr 6905 df-c 6987 df-add 6992 |
This theorem is referenced by: addresr 7005 addcnsrec 7010 axaddcl 7032 axaddcom 7036 ax0id 7044 axcnre 7047 |
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