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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 10635 | . . 3 ⊢ R = ((P × P) / ~R ) | |
2 | mulsrpr 10655 | . . 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 10655 | . . 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 10602 | . . . 4 ⊢ (𝑥 ·P 𝑧) = (𝑧 ·P 𝑥) | |
5 | mulcompr 10602 | . . . 4 ⊢ (𝑦 ·P 𝑤) = (𝑤 ·P 𝑦) | |
6 | 4, 5 | oveq12i 7203 | . . 3 ⊢ ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = ((𝑧 ·P 𝑥) +P (𝑤 ·P 𝑦)) |
7 | mulcompr 10602 | . . . . 5 ⊢ (𝑥 ·P 𝑤) = (𝑤 ·P 𝑥) | |
8 | mulcompr 10602 | . . . . 5 ⊢ (𝑦 ·P 𝑧) = (𝑧 ·P 𝑦) | |
9 | 7, 8 | oveq12i 7203 | . . . 4 ⊢ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) = ((𝑤 ·P 𝑥) +P (𝑧 ·P 𝑦)) |
10 | addcompr 10600 | . . . 4 ⊢ ((𝑤 ·P 𝑥) +P (𝑧 ·P 𝑦)) = ((𝑧 ·P 𝑦) +P (𝑤 ·P 𝑥)) | |
11 | 9, 10 | eqtri 2759 | . . 3 ⊢ ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) = ((𝑧 ·P 𝑦) +P (𝑤 ·P 𝑥)) |
12 | 1, 2, 3, 6, 11 | ecovcom 8483 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴)) |
13 | dmmulsr 10665 | . . 3 ⊢ dom ·R = (R × R) | |
14 | 13 | ndmovcom 7373 | . 2 ⊢ (¬ (𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴)) |
15 | 12, 14 | pm2.61i 185 | 1 ⊢ (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2112 (class class class)co 7191 Pcnp 10438 +P cpp 10440 ·P cmp 10441 ~R cer 10443 Rcnr 10444 ·R cmr 10449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-omul 8185 df-er 8369 df-ec 8371 df-qs 8375 df-ni 10451 df-pli 10452 df-mi 10453 df-lti 10454 df-plpq 10487 df-mpq 10488 df-ltpq 10489 df-enq 10490 df-nq 10491 df-erq 10492 df-plq 10493 df-mq 10494 df-1nq 10495 df-rq 10496 df-ltnq 10497 df-np 10560 df-plp 10562 df-mp 10563 df-ltp 10564 df-enr 10634 df-nr 10635 df-mr 10637 |
This theorem is referenced by: sqgt0sr 10685 mulresr 10718 axmulcom 10734 axmulass 10736 axcnre 10743 |
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