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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 10212 be used later. Instead, use mulcom 10234. (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 10175 | . 2 ⊢ ℂ = ((R × R) / ◡ E ) | |
2 | mulcnsrec 10177 | . 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 10177 | . 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 10122 | . . 3 ⊢ (𝑥 ·R 𝑧) = (𝑧 ·R 𝑥) | |
5 | mulcomsr 10122 | . . . 4 ⊢ (𝑦 ·R 𝑤) = (𝑤 ·R 𝑦) | |
6 | 5 | oveq2i 6825 | . . 3 ⊢ (-1R ·R (𝑦 ·R 𝑤)) = (-1R ·R (𝑤 ·R 𝑦)) |
7 | 4, 6 | oveq12i 6826 | . 2 ⊢ ((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))) = ((𝑧 ·R 𝑥) +R (-1R ·R (𝑤 ·R 𝑦))) |
8 | mulcomsr 10122 | . . . 4 ⊢ (𝑦 ·R 𝑧) = (𝑧 ·R 𝑦) | |
9 | mulcomsr 10122 | . . . 4 ⊢ (𝑥 ·R 𝑤) = (𝑤 ·R 𝑥) | |
10 | 8, 9 | oveq12i 6826 | . . 3 ⊢ ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) = ((𝑧 ·R 𝑦) +R (𝑤 ·R 𝑥)) |
11 | addcomsr 10120 | . . 3 ⊢ ((𝑧 ·R 𝑦) +R (𝑤 ·R 𝑥)) = ((𝑤 ·R 𝑥) +R (𝑧 ·R 𝑦)) | |
12 | 10, 11 | eqtri 2782 | . 2 ⊢ ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) = ((𝑤 ·R 𝑥) +R (𝑧 ·R 𝑦)) |
13 | 1, 2, 3, 7, 12 | ecovcom 8022 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 E cep 5178 ◡ccnv 5265 (class class class)co 6814 Rcnr 9899 -1Rcm1r 9902 +R cplr 9903 ·R cmr 9904 ℂcc 10146 · cmul 10153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-omul 7735 df-er 7913 df-ec 7915 df-qs 7919 df-ni 9906 df-pli 9907 df-mi 9908 df-lti 9909 df-plpq 9942 df-mpq 9943 df-ltpq 9944 df-enq 9945 df-nq 9946 df-erq 9947 df-plq 9948 df-mq 9949 df-1nq 9950 df-rq 9951 df-ltnq 9952 df-np 10015 df-plp 10017 df-mp 10018 df-ltp 10019 df-enr 10089 df-nr 10090 df-plr 10091 df-mr 10092 df-c 10154 df-mul 10160 |
This theorem is referenced by: (None) |
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