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Mirrors > Home > ILE Home > Th. List > mulcomnqg | GIF version |
Description: Multiplication of positive fractions is commutative. (Contributed by Jim Kingdon, 17-Sep-2019.) |
Ref | Expression |
---|---|
mulcomnqg | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 7124 | . 2 ⊢ Q = ((N × N) / ~Q ) | |
2 | mulpipqqs 7149 | . 2 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → ([〈𝑥, 𝑦〉] ~Q ·Q [〈𝑧, 𝑤〉] ~Q ) = [〈(𝑥 ·N 𝑧), (𝑦 ·N 𝑤)〉] ~Q ) | |
3 | mulpipqqs 7149 | . 2 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑥 ∈ N ∧ 𝑦 ∈ N)) → ([〈𝑧, 𝑤〉] ~Q ·Q [〈𝑥, 𝑦〉] ~Q ) = [〈(𝑧 ·N 𝑥), (𝑤 ·N 𝑦)〉] ~Q ) | |
4 | mulcompig 7107 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥)) | |
5 | 4 | ad2ant2r 500 | . 2 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥)) |
6 | mulcompig 7107 | . . 3 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·N 𝑤) = (𝑤 ·N 𝑦)) | |
7 | 6 | ad2ant2l 499 | . 2 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → (𝑦 ·N 𝑤) = (𝑤 ·N 𝑦)) |
8 | 1, 2, 3, 5, 7 | ecovicom 6505 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 (class class class)co 5742 Ncnpi 7048 ·N cmi 7050 ~Q ceq 7055 Qcnq 7056 ·Q cmq 7059 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-mi 7082 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-mqqs 7126 |
This theorem is referenced by: recmulnqg 7167 recrecnq 7170 rec1nq 7171 lt2mulnq 7181 halfnqq 7186 prarloclemarch 7194 prarloclemarch2 7195 ltrnqg 7196 prarloclemlt 7269 addnqprllem 7303 addnqprulem 7304 addnqprl 7305 addnqpru 7306 appdivnq 7339 prmuloclemcalc 7341 mulnqprl 7344 mulnqpru 7345 mullocprlem 7346 mulclpr 7348 mulcomprg 7356 distrlem4prl 7360 distrlem4pru 7361 1idprl 7366 1idpru 7367 recexprlem1ssl 7409 recexprlem1ssu 7410 recexprlemss1l 7411 recexprlemss1u 7412 |
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