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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 7303 | . 2 ⊢ Q = ((N × N) / ~Q ) | |
2 | mulpipqqs 7328 | . 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 7328 | . 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 7286 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥)) | |
5 | 4 | ad2ant2r 506 | . 2 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥)) |
6 | mulcompig 7286 | . . 3 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·N 𝑤) = (𝑤 ·N 𝑦)) | |
7 | 6 | ad2ant2l 505 | . 2 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → (𝑦 ·N 𝑤) = (𝑤 ·N 𝑦)) |
8 | 1, 2, 3, 5, 7 | ecovicom 6619 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 (class class class)co 5851 Ncnpi 7227 ·N cmi 7229 ~Q ceq 7234 Qcnq 7235 ·Q cmq 7238 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-oadd 6397 df-omul 6398 df-er 6511 df-ec 6513 df-qs 6517 df-ni 7259 df-mi 7261 df-mpq 7300 df-enq 7302 df-nqqs 7303 df-mqqs 7305 |
This theorem is referenced by: recmulnqg 7346 recrecnq 7349 rec1nq 7350 lt2mulnq 7360 halfnqq 7365 prarloclemarch 7373 prarloclemarch2 7374 ltrnqg 7375 prarloclemlt 7448 addnqprllem 7482 addnqprulem 7483 addnqprl 7484 addnqpru 7485 appdivnq 7518 prmuloclemcalc 7520 mulnqprl 7523 mulnqpru 7524 mullocprlem 7525 mulclpr 7527 mulcomprg 7535 distrlem4prl 7539 distrlem4pru 7540 1idprl 7545 1idpru 7546 recexprlem1ssl 7588 recexprlem1ssu 7589 recexprlemss1l 7590 recexprlemss1u 7591 |
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