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Mirrors > Home > MPE Home > Th. List > mulcomsr | Structured version Visualization version GIF version |
Description: Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulcomsr | ⊢ (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 10466 | . . 3 ⊢ R = ((P × P) / ~R ) | |
2 | mulsrpr 10486 | . . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([〈𝑥, 𝑦〉] ~R ·R [〈𝑧, 𝑤〉] ~R ) = [〈((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)), ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧))〉] ~R ) | |
3 | mulsrpr 10486 | . . 3 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → ([〈𝑧, 𝑤〉] ~R ·R [〈𝑥, 𝑦〉] ~R ) = [〈((𝑧 ·P 𝑥) +P (𝑤 ·P 𝑦)), ((𝑧 ·P 𝑦) +P (𝑤 ·P 𝑥))〉] ~R ) | |
4 | mulcompr 10433 | . . . 4 ⊢ (𝑥 ·P 𝑧) = (𝑧 ·P 𝑥) | |
5 | mulcompr 10433 | . . . 4 ⊢ (𝑦 ·P 𝑤) = (𝑤 ·P 𝑦) | |
6 | 4, 5 | oveq12i 7157 | . . 3 ⊢ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = ((𝑧 ·P 𝑥) +P (𝑤 ·P 𝑦)) |
7 | mulcompr 10433 | . . . . 5 ⊢ (𝑥 ·P 𝑤) = (𝑤 ·P 𝑥) | |
8 | mulcompr 10433 | . . . . 5 ⊢ (𝑦 ·P 𝑧) = (𝑧 ·P 𝑦) | |
9 | 7, 8 | oveq12i 7157 | . . . 4 ⊢ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) = ((𝑤 ·P 𝑥) +P (𝑧 ·P 𝑦)) |
10 | addcompr 10431 | . . . 4 ⊢ ((𝑤 ·P 𝑥) +P (𝑧 ·P 𝑦)) = ((𝑧 ·P 𝑦) +P (𝑤 ·P 𝑥)) | |
11 | 9, 10 | eqtri 2841 | . . 3 ⊢ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) = ((𝑧 ·P 𝑦) +P (𝑤 ·P 𝑥)) |
12 | 1, 2, 3, 6, 11 | ecovcom 8392 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴)) |
13 | dmmulsr 10496 | . . 3 ⊢ dom ·R = (R × R) | |
14 | 13 | ndmovcom 7324 | . 2 ⊢ (¬ (𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴)) |
15 | 12, 14 | pm2.61i 183 | 1 ⊢ (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 Pcnp 10269 +P cpp 10271 ·P cmp 10272 ~R cer 10274 Rcnr 10275 ·R cmr 10280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-omul 8096 df-er 8278 df-ec 8280 df-qs 8284 df-ni 10282 df-pli 10283 df-mi 10284 df-lti 10285 df-plpq 10318 df-mpq 10319 df-ltpq 10320 df-enq 10321 df-nq 10322 df-erq 10323 df-plq 10324 df-mq 10325 df-1nq 10326 df-rq 10327 df-ltnq 10328 df-np 10391 df-plp 10393 df-mp 10394 df-ltp 10395 df-enr 10465 df-nr 10466 df-mr 10468 |
This theorem is referenced by: sqgt0sr 10516 mulresr 10549 axmulcom 10565 axmulass 10567 axcnre 10574 |
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