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Mirrors > Home > MPE Home > Th. List > mulcomnq | Structured version Visualization version GIF version |
Description: Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulcomnq | ⊢ (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcompq 10884 | . . . 4 ⊢ (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴) | |
2 | 1 | fveq2i 6842 | . . 3 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘(𝐵 ·pQ 𝐴)) |
3 | mulpqnq 10873 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) | |
4 | mulpqnq 10873 | . . . 4 ⊢ ((𝐵 ∈ Q ∧ 𝐴 ∈ Q) → (𝐵 ·Q 𝐴) = ([Q]‘(𝐵 ·pQ 𝐴))) | |
5 | 4 | ancoms 459 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 ·Q 𝐴) = ([Q]‘(𝐵 ·pQ 𝐴))) |
6 | 2, 3, 5 | 3eqtr4a 2802 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)) |
7 | mulnqf 10881 | . . . 4 ⊢ ·Q :(Q × Q)⟶Q | |
8 | 7 | fdmi 6677 | . . 3 ⊢ dom ·Q = (Q × Q) |
9 | 8 | ndmovcom 7537 | . 2 ⊢ (¬ (𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)) |
10 | 6, 9 | pm2.61i 182 | 1 ⊢ (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 × cxp 5629 ‘cfv 6493 (class class class)co 7353 ·pQ cmpq 10781 Qcnq 10784 [Q]cerq 10786 ·Q cmq 10788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-oadd 8412 df-omul 8413 df-er 8644 df-ni 10804 df-mi 10806 df-lti 10807 df-mpq 10841 df-enq 10843 df-nq 10844 df-erq 10845 df-mq 10847 df-1nq 10848 |
This theorem is referenced by: recmulnq 10896 recrecnq 10899 halfnq 10908 ltrnq 10911 addclprlem1 10948 addclprlem2 10949 mulclprlem 10951 mulclpr 10952 mulcompr 10955 distrlem4pr 10958 1idpr 10961 prlem934 10965 prlem936 10979 reclem3pr 10981 reclem4pr 10982 |
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