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Mirrors > Home > MPE Home > Th. List > addcompq | 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 |
---|---|
addcompq | ⊢ (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcompi 10310 | . . . 4 ⊢ (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))) = (((1st ‘𝐵) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐵))) | |
2 | mulcompi 10312 | . . . 4 ⊢ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) = ((2nd ‘𝐵) ·N (2nd ‘𝐴)) | |
3 | 1, 2 | opeq12i 4801 | . . 3 ⊢ 〈(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉 = 〈(((1st ‘𝐵) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐴))〉 |
4 | addpipq2 10352 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = 〈(((1st ‘𝐴) ·N (2nd ‘𝐵)) +N ((1st ‘𝐵) ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N (2nd ‘𝐵))〉) | |
5 | addpipq2 10352 | . . . 4 ⊢ ((𝐵 ∈ (N × N) ∧ 𝐴 ∈ (N × N)) → (𝐵 +pQ 𝐴) = 〈(((1st ‘𝐵) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐴))〉) | |
6 | 5 | ancoms 461 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐵 +pQ 𝐴) = 〈(((1st ‘𝐵) ·N (2nd ‘𝐴)) +N ((1st ‘𝐴) ·N (2nd ‘𝐵))), ((2nd ‘𝐵) ·N (2nd ‘𝐴))〉) |
7 | 3, 4, 6 | 3eqtr4a 2882 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴)) |
8 | addpqf 10360 | . . . 4 ⊢ +pQ :((N × N) × (N × N))⟶(N × N) | |
9 | 8 | fdmi 6518 | . . 3 ⊢ dom +pQ = ((N × N) × (N × N)) |
10 | 9 | ndmovcom 7329 | . 2 ⊢ (¬ (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴)) |
11 | 7, 10 | pm2.61i 184 | 1 ⊢ (𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 〈cop 4566 × cxp 5547 ‘cfv 6349 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 Ncnpi 10260 +N cpli 10261 ·N cmi 10262 +pQ cplpq 10264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-oadd 8100 df-omul 8101 df-ni 10288 df-pli 10289 df-mi 10290 df-plpq 10324 |
This theorem is referenced by: addcomnq 10367 adderpq 10372 |
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