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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 9774 | . . . 4 ⊢ (𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴) | |
2 | 1 | fveq2i 6194 | . . 3 ⊢ ([Q]‘(𝐴 ·pQ 𝐵)) = ([Q]‘(𝐵 ·pQ 𝐴)) |
3 | mulpqnq 9763 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵))) | |
4 | mulpqnq 9763 | . . . 4 ⊢ ((𝐵 ∈ Q ∧ 𝐴 ∈ Q) → (𝐵 ·Q 𝐴) = ([Q]‘(𝐵 ·pQ 𝐴))) | |
5 | 4 | ancoms 469 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 ·Q 𝐴) = ([Q]‘(𝐵 ·pQ 𝐴))) |
6 | 2, 3, 5 | 3eqtr4a 2682 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)) |
7 | mulnqf 9771 | . . . 4 ⊢ ·Q :(Q × Q)⟶Q | |
8 | 7 | fdmi 6052 | . . 3 ⊢ dom ·Q = (Q × Q) |
9 | 8 | ndmovcom 6821 | . 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 384 = wceq 1483 ∈ wcel 1990 × cxp 5112 ‘cfv 5888 (class class class)co 6650 ·pQ cmpq 9671 Qcnq 9674 [Q]cerq 9676 ·Q cmq 9678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ni 9694 df-mi 9696 df-lti 9697 df-mpq 9731 df-enq 9733 df-nq 9734 df-erq 9735 df-mq 9737 df-1nq 9738 |
This theorem is referenced by: recmulnq 9786 recrecnq 9789 halfnq 9798 ltrnq 9801 addclprlem1 9838 addclprlem2 9839 mulclprlem 9841 mulclpr 9842 mulcompr 9845 distrlem4pr 9848 1idpr 9851 prlem934 9855 prlem936 9869 reclem3pr 9871 reclem4pr 9872 |
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