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Mirrors > Home > ILE Home > Th. List > mulcomsrg | GIF version |
Description: Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Ref | Expression |
---|---|
mulcomsrg | ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 7559 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | mulsrpr 7578 | . 2 ⊢ (((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 7578 | . 2 ⊢ (((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 | mulcomprg 7412 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 ·P 𝑧) = (𝑧 ·P 𝑥)) | |
5 | 4 | ad2ant2r 501 | . . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑥 ·P 𝑧) = (𝑧 ·P 𝑥)) |
6 | mulcomprg 7412 | . . . 4 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 ·P 𝑤) = (𝑤 ·P 𝑦)) | |
7 | 6 | ad2ant2l 500 | . . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦 ·P 𝑤) = (𝑤 ·P 𝑦)) |
8 | 5, 7 | oveq12d 5800 | . 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑧) +P (𝑦 ·P 𝑤)) = ((𝑧 ·P 𝑥) +P (𝑤 ·P 𝑦))) |
9 | mulcomprg 7412 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑥 ·P 𝑤) = (𝑤 ·P 𝑥)) | |
10 | 9 | ad2ant2rl 503 | . . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑥 ·P 𝑤) = (𝑤 ·P 𝑥)) |
11 | mulcomprg 7412 | . . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑦 ·P 𝑧) = (𝑧 ·P 𝑦)) | |
12 | 11 | ad2ant2lr 502 | . . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦 ·P 𝑧) = (𝑧 ·P 𝑦)) |
13 | 10, 12 | oveq12d 5800 | . . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) = ((𝑤 ·P 𝑥) +P (𝑧 ·P 𝑦))) |
14 | mulclpr 7404 | . . . . . 6 ⊢ ((𝑤 ∈ P ∧ 𝑥 ∈ P) → (𝑤 ·P 𝑥) ∈ P) | |
15 | 14 | ancoms 266 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑤 ∈ P) → (𝑤 ·P 𝑥) ∈ P) |
16 | 15 | ad2ant2rl 503 | . . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑤 ·P 𝑥) ∈ P) |
17 | mulclpr 7404 | . . . . . 6 ⊢ ((𝑧 ∈ P ∧ 𝑦 ∈ P) → (𝑧 ·P 𝑦) ∈ P) | |
18 | 17 | ancoms 266 | . . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑧 ·P 𝑦) ∈ P) |
19 | 18 | ad2ant2lr 502 | . . . 4 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑧 ·P 𝑦) ∈ P) |
20 | addcomprg 7410 | . . . 4 ⊢ (((𝑤 ·P 𝑥) ∈ P ∧ (𝑧 ·P 𝑦) ∈ P) → ((𝑤 ·P 𝑥) +P (𝑧 ·P 𝑦)) = ((𝑧 ·P 𝑦) +P (𝑤 ·P 𝑥))) | |
21 | 16, 19, 20 | syl2anc 409 | . . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑤 ·P 𝑥) +P (𝑧 ·P 𝑦)) = ((𝑧 ·P 𝑦) +P (𝑤 ·P 𝑥))) |
22 | 13, 21 | eqtrd 2173 | . 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ((𝑥 ·P 𝑤) +P (𝑦 ·P 𝑧)) = ((𝑧 ·P 𝑦) +P (𝑤 ·P 𝑥))) |
23 | 1, 2, 3, 8, 22 | ecovicom 6545 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 (class class class)co 5782 Pcnp 7123 +P cpp 7125 ·P cmp 7126 ~R cer 7128 Rcnr 7129 ·R cmr 7134 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-iplp 7300 df-imp 7301 df-enr 7558 df-nr 7559 df-mr 7561 |
This theorem is referenced by: mulresr 7670 axmulcom 7703 axmulass 7705 axcnre 7713 |
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