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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 7376 | . 2 ⊢ Q = ((N × N) / ~Q ) | |
2 | mulpipqqs 7401 | . 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 7401 | . 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 7359 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥)) | |
5 | 4 | ad2ant2r 509 | . 2 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥)) |
6 | mulcompig 7359 | . . 3 ⊢ ((𝑦 ∈ N ∧ 𝑤 ∈ N) → (𝑦 ·N 𝑤) = (𝑤 ·N 𝑦)) | |
7 | 6 | ad2ant2l 508 | . 2 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑤 ∈ N)) → (𝑦 ·N 𝑤) = (𝑤 ·N 𝑦)) |
8 | 1, 2, 3, 5, 7 | ecovicom 6668 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 (class class class)co 5895 Ncnpi 7300 ·N cmi 7302 ~Q ceq 7307 Qcnq 7308 ·Q cmq 7311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-recs 6329 df-irdg 6394 df-oadd 6444 df-omul 6445 df-er 6558 df-ec 6560 df-qs 6564 df-ni 7332 df-mi 7334 df-mpq 7373 df-enq 7375 df-nqqs 7376 df-mqqs 7378 |
This theorem is referenced by: recmulnqg 7419 recrecnq 7422 rec1nq 7423 lt2mulnq 7433 halfnqq 7438 prarloclemarch 7446 prarloclemarch2 7447 ltrnqg 7448 prarloclemlt 7521 addnqprllem 7555 addnqprulem 7556 addnqprl 7557 addnqpru 7558 appdivnq 7591 prmuloclemcalc 7593 mulnqprl 7596 mulnqpru 7597 mullocprlem 7598 mulclpr 7600 mulcomprg 7608 distrlem4prl 7612 distrlem4pru 7613 1idprl 7618 1idpru 7619 recexprlem1ssl 7661 recexprlem1ssu 7662 recexprlemss1l 7663 recexprlemss1u 7664 |
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