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Mirrors > Home > MPE Home > Th. List > axmulcom | Structured version Visualization version GIF version |
Description: Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 10590 be used later. Instead, use mulcom 10612. (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 10553 | . 2 ⊢ ℂ = ((R × R) / ◡ E ) | |
2 | mulcnsrec 10555 | . 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 10555 | . 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 | mulcomsr 10500 | . . 3 ⊢ (𝑥 ·R 𝑧) = (𝑧 ·R 𝑥) | |
5 | mulcomsr 10500 | . . . 4 ⊢ (𝑦 ·R 𝑤) = (𝑤 ·R 𝑦) | |
6 | 5 | oveq2i 7146 | . . 3 ⊢ (-1R ·R (𝑦 ·R 𝑤)) = (-1R ·R (𝑤 ·R 𝑦)) |
7 | 4, 6 | oveq12i 7147 | . 2 ⊢ ((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))) = ((𝑧 ·R 𝑥) +R (-1R ·R (𝑤 ·R 𝑦))) |
8 | mulcomsr 10500 | . . . 4 ⊢ (𝑦 ·R 𝑧) = (𝑧 ·R 𝑦) | |
9 | mulcomsr 10500 | . . . 4 ⊢ (𝑥 ·R 𝑤) = (𝑤 ·R 𝑥) | |
10 | 8, 9 | oveq12i 7147 | . . 3 ⊢ ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) = ((𝑧 ·R 𝑦) +R (𝑤 ·R 𝑥)) |
11 | addcomsr 10498 | . . 3 ⊢ ((𝑧 ·R 𝑦) +R (𝑤 ·R 𝑥)) = ((𝑤 ·R 𝑥) +R (𝑧 ·R 𝑦)) | |
12 | 10, 11 | eqtri 2821 | . 2 ⊢ ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) = ((𝑤 ·R 𝑥) +R (𝑧 ·R 𝑦)) |
13 | 1, 2, 3, 7, 12 | ecovcom 8386 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 E cep 5429 ◡ccnv 5518 (class class class)co 7135 Rcnr 10276 -1Rcm1r 10279 +R cplr 10280 ·R cmr 10281 ℂcc 10524 · cmul 10531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-ec 8274 df-qs 8278 df-ni 10283 df-pli 10284 df-mi 10285 df-lti 10286 df-plpq 10319 df-mpq 10320 df-ltpq 10321 df-enq 10322 df-nq 10323 df-erq 10324 df-plq 10325 df-mq 10326 df-1nq 10327 df-rq 10328 df-ltnq 10329 df-np 10392 df-plp 10394 df-mp 10395 df-ltp 10396 df-enr 10466 df-nr 10467 df-plr 10468 df-mr 10469 df-c 10532 df-mul 10538 |
This theorem is referenced by: (None) |
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