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Mirrors > Home > MPE Home > Th. List > addcomnq | Structured version Visualization version GIF version |
Description: Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addcomnq | ⊢ (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcompq 10944 | . . . 4 ⊢ (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴) | |
2 | 1 | fveq2i 6894 | . . 3 ⊢ ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘(𝐵 +pQ 𝐴)) |
3 | addpqnq 10932 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵))) | |
4 | addpqnq 10932 | . . . 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 2798 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴)) |
7 | addnqf 10942 | . . . 4 ⊢ +Q :(Q × Q)⟶Q | |
8 | 7 | fdmi 6729 | . . 3 ⊢ dom +Q = (Q × Q) |
9 | 8 | ndmovcom 7593 | . 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 5674 ‘cfv 6543 (class class class)co 7408 +pQ cplpq 10842 Qcnq 10846 [Q]cerq 10848 +Q cplq 10849 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-oadd 8469 df-omul 8470 df-er 8702 df-ni 10866 df-pli 10867 df-mi 10868 df-lti 10869 df-plpq 10902 df-enq 10905 df-nq 10906 df-erq 10907 df-plq 10908 df-1nq 10910 |
This theorem is referenced by: ltaddnq 10968 addclprlem2 11011 addclpr 11012 addcompr 11015 distrlem4pr 11020 prlem934 11027 ltexprlem2 11031 ltexprlem6 11035 ltexprlem7 11036 prlem936 11041 |
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