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Mirrors > Home > ILE Home > Th. List > axmulcom | GIF version |
Description: Multiplication of complex numbers is commutative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 7714 be used later. Instead, use mulcom 7742. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axmulcom | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcnqs 7642 | . 2 ⊢ ℂ = ((R × R) / ◡ E ) | |
2 | mulcnsrec 7644 | . 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ([〈𝑥, 𝑦〉]◡ E · [〈𝑧, 𝑤〉]◡ E ) = [〈((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))〉]◡ E ) | |
3 | mulcnsrec 7644 | . 2 ⊢ (((𝑧 ∈ R ∧ 𝑤 ∈ R) ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → ([〈𝑧, 𝑤〉]◡ E · [〈𝑥, 𝑦〉]◡ E ) = [〈((𝑧 ·R 𝑥) +R (-1R ·R (𝑤 ·R 𝑦))), ((𝑤 ·R 𝑥) +R (𝑧 ·R 𝑦))〉]◡ E ) | |
4 | simpll 518 | . . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → 𝑥 ∈ R) | |
5 | simprl 520 | . . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → 𝑧 ∈ R) | |
6 | mulcomsrg 7558 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 ·R 𝑧) = (𝑧 ·R 𝑥)) | |
7 | 4, 5, 6 | syl2anc 408 | . . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑥 ·R 𝑧) = (𝑧 ·R 𝑥)) |
8 | simplr 519 | . . . . 5 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → 𝑦 ∈ R) | |
9 | simprr 521 | . . . . 5 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → 𝑤 ∈ R) | |
10 | mulcomsrg 7558 | . . . . 5 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 ·R 𝑤) = (𝑤 ·R 𝑦)) | |
11 | 8, 9, 10 | syl2anc 408 | . . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑦 ·R 𝑤) = (𝑤 ·R 𝑦)) |
12 | 11 | oveq2d 5783 | . . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (-1R ·R (𝑦 ·R 𝑤)) = (-1R ·R (𝑤 ·R 𝑦))) |
13 | 7, 12 | oveq12d 5785 | . 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))) = ((𝑧 ·R 𝑥) +R (-1R ·R (𝑤 ·R 𝑦)))) |
14 | mulcomsrg 7558 | . . . . 5 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑦 ·R 𝑧) = (𝑧 ·R 𝑦)) | |
15 | 8, 5, 14 | syl2anc 408 | . . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑦 ·R 𝑧) = (𝑧 ·R 𝑦)) |
16 | mulcomsrg 7558 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑤 ∈ R) → (𝑥 ·R 𝑤) = (𝑤 ·R 𝑥)) | |
17 | 4, 9, 16 | syl2anc 408 | . . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑥 ·R 𝑤) = (𝑤 ·R 𝑥)) |
18 | 15, 17 | oveq12d 5785 | . . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) = ((𝑧 ·R 𝑦) +R (𝑤 ·R 𝑥))) |
19 | mulclsr 7555 | . . . . 5 ⊢ ((𝑧 ∈ R ∧ 𝑦 ∈ R) → (𝑧 ·R 𝑦) ∈ R) | |
20 | 5, 8, 19 | syl2anc 408 | . . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑧 ·R 𝑦) ∈ R) |
21 | mulclsr 7555 | . . . . 5 ⊢ ((𝑤 ∈ R ∧ 𝑥 ∈ R) → (𝑤 ·R 𝑥) ∈ R) | |
22 | 9, 4, 21 | syl2anc 408 | . . . 4 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (𝑤 ·R 𝑥) ∈ R) |
23 | addcomsrg 7556 | . . . 4 ⊢ (((𝑧 ·R 𝑦) ∈ R ∧ (𝑤 ·R 𝑥) ∈ R) → ((𝑧 ·R 𝑦) +R (𝑤 ·R 𝑥)) = ((𝑤 ·R 𝑥) +R (𝑧 ·R 𝑦))) | |
24 | 20, 22, 23 | syl2anc 408 | . . 3 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ((𝑧 ·R 𝑦) +R (𝑤 ·R 𝑥)) = ((𝑤 ·R 𝑥) +R (𝑧 ·R 𝑦))) |
25 | 18, 24 | eqtrd 2170 | . 2 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) = ((𝑤 ·R 𝑥) +R (𝑧 ·R 𝑦))) |
26 | 1, 2, 3, 13, 25 | ecovicom 6530 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 E cep 4204 ◡ccnv 4533 (class class class)co 5767 Rcnr 7098 -1Rcm1r 7101 +R cplr 7102 ·R cmr 7103 ℂcc 7611 · cmul 7618 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-2o 6307 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-enq0 7225 df-nq0 7226 df-0nq0 7227 df-plq0 7228 df-mq0 7229 df-inp 7267 df-i1p 7268 df-iplp 7269 df-imp 7270 df-enr 7527 df-nr 7528 df-plr 7529 df-mr 7530 df-m1r 7534 df-c 7619 df-mul 7625 |
This theorem is referenced by: rereceu 7690 recriota 7691 |
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