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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 11015 | . . . 4 ⊢ (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴) | |
2 | 1 | fveq2i 6922 | . . 3 ⊢ ([Q]‘(𝐴 +pQ 𝐵)) = ([Q]‘(𝐵 +pQ 𝐴)) |
3 | addpqnq 11003 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵))) | |
4 | addpqnq 11003 | . . . 4 ⊢ ((𝐵 ∈ Q ∧ 𝐴 ∈ Q) → (𝐵 +Q 𝐴) = ([Q]‘(𝐵 +pQ 𝐴))) | |
5 | 4 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 +Q 𝐴) = ([Q]‘(𝐵 +pQ 𝐴))) |
6 | 2, 3, 5 | 3eqtr4a 2800 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴)) |
7 | addnqf 11013 | . . . 4 ⊢ +Q :(Q × Q)⟶Q | |
8 | 7 | fdmi 6757 | . . 3 ⊢ dom +Q = (Q × Q) |
9 | 8 | ndmovcom 7633 | . 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 395 = wceq 1537 ∈ wcel 2103 × cxp 5697 ‘cfv 6572 (class class class)co 7445 +pQ cplpq 10913 Qcnq 10917 [Q]cerq 10919 +Q cplq 10920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-oadd 8522 df-omul 8523 df-er 8759 df-ni 10937 df-pli 10938 df-mi 10939 df-lti 10940 df-plpq 10973 df-enq 10976 df-nq 10977 df-erq 10978 df-plq 10979 df-1nq 10981 |
This theorem is referenced by: ltaddnq 11039 addclprlem2 11082 addclpr 11083 addcompr 11086 distrlem4pr 11091 prlem934 11098 ltexprlem2 11102 ltexprlem6 11106 ltexprlem7 11107 prlem936 11112 |
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