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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 9964 | . . . 4 ⊢ (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴) | |
2 | 1 | fveq2i 6353 | . . 3 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘(𝐵 ·pQ 𝐴)) |
3 | mulpqnq 9953 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) | |
4 | mulpqnq 9953 | . . . 4 ⊢ ((𝐵 ∈ Q ∧ 𝐴 ∈ Q) → (𝐵 ·Q 𝐴) = ([Q]‘(𝐵 ·pQ 𝐴))) | |
5 | 4 | ancoms 468 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 ·Q 𝐴) = ([Q]‘(𝐵 ·pQ 𝐴))) |
6 | 2, 3, 5 | 3eqtr4a 2818 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)) |
7 | mulnqf 9961 | . . . 4 ⊢ ·Q :(Q × Q)⟶Q | |
8 | 7 | fdmi 6211 | . . 3 ⊢ dom ·Q = (Q × Q) |
9 | 8 | ndmovcom 6984 | . 2 ⊢ (¬ (𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)) |
10 | 6, 9 | pm2.61i 176 | 1 ⊢ (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1630 ∈ wcel 2137 × cxp 5262 ‘cfv 6047 (class class class)co 6811 ·pQ cmpq 9861 Qcnq 9864 [Q]cerq 9866 ·Q cmq 9868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-ral 3053 df-rex 3054 df-reu 3055 df-rmo 3056 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-pss 3729 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-tp 4324 df-op 4326 df-uni 4587 df-iun 4672 df-br 4803 df-opab 4863 df-mpt 4880 df-tr 4903 df-id 5172 df-eprel 5177 df-po 5185 df-so 5186 df-fr 5223 df-we 5225 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-pred 5839 df-ord 5885 df-on 5886 df-lim 5887 df-suc 5888 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-om 7229 df-1st 7331 df-2nd 7332 df-wrecs 7574 df-recs 7635 df-rdg 7673 df-1o 7727 df-oadd 7731 df-omul 7732 df-er 7909 df-ni 9884 df-mi 9886 df-lti 9887 df-mpq 9921 df-enq 9923 df-nq 9924 df-erq 9925 df-mq 9927 df-1nq 9928 |
This theorem is referenced by: recmulnq 9976 recrecnq 9979 halfnq 9988 ltrnq 9991 addclprlem1 10028 addclprlem2 10029 mulclprlem 10031 mulclpr 10032 mulcompr 10035 distrlem4pr 10038 1idpr 10041 prlem934 10045 prlem936 10059 reclem3pr 10061 reclem4pr 10062 |
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