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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 6332 | . 2 ⊢ Q = ((N × N) / ~Q ) | |
2 | mulpipqqs 6357 | . 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 6357 | . 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 6315 | . . 3 ⊢ ((x ∈ N ∧ z ∈ N) → (x ·N z) = (z ·N x)) | |
5 | 4 | ad2ant2r 478 | . 2 ⊢ (((x ∈ N ∧ y ∈ N) ∧ (z ∈ N ∧ w ∈ N)) → (x ·N z) = (z ·N x)) |
6 | mulcompig 6315 | . . 3 ⊢ ((y ∈ N ∧ w ∈ N) → (y ·N w) = (w ·N y)) | |
7 | 6 | ad2ant2l 477 | . 2 ⊢ (((x ∈ N ∧ y ∈ N) ∧ (z ∈ N ∧ w ∈ N)) → (y ·N w) = (w ·N y)) |
8 | 1, 2, 3, 5, 7 | ecovicom 6150 | 1 ⊢ ((A ∈ Q ∧ B ∈ Q) → (A ·Q B) = (B ·Q A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 (class class class)co 5455 Ncnpi 6256 ·N cmi 6258 ~Q ceq 6263 Qcnq 6264 ·Q cmq 6267 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-mi 6290 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-mqqs 6334 |
This theorem is referenced by: recmulnqg 6375 recrecnq 6378 rec1nq 6379 halfnqq 6393 prarloclemarch 6401 prarloclemarch2 6402 ltrnqg 6403 prarloclemlt 6476 addnqprllem 6510 addnqprulem 6511 addnqprl 6512 addnqpru 6513 appdivnq 6544 prmuloclemcalc 6546 mulnqprl 6549 mulnqpru 6550 mullocprlem 6551 mulclpr 6553 mulcomprg 6556 distrlem4prl 6560 distrlem4pru 6561 1idprl 6566 1idpru 6567 recexprlem1ssl 6605 recexprlem1ssu 6606 recexprlemss1l 6607 recexprlemss1u 6608 |
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