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Mirrors > Home > MPE Home > Th. List > addcomsr | Structured version Visualization version GIF version |
Description: Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addcomsr | ⊢ (𝐴 +R 𝐵) = (𝐵 +R 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 10480 | . . 3 ⊢ R = ((P × P) / ~R ) | |
2 | addsrpr 10499 | . . 3 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([〈𝑥, 𝑦〉] ~R +R [〈𝑧, 𝑤〉] ~R ) = [〈(𝑥 +P 𝑧), (𝑦 +P 𝑤)〉] ~R ) | |
3 | addsrpr 10499 | . . 3 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → ([〈𝑧, 𝑤〉] ~R +R [〈𝑥, 𝑦〉] ~R ) = [〈(𝑧 +P 𝑥), (𝑤 +P 𝑦)〉] ~R ) | |
4 | addcompr 10445 | . . 3 ⊢ (𝑥 +P 𝑧) = (𝑧 +P 𝑥) | |
5 | addcompr 10445 | . . 3 ⊢ (𝑦 +P 𝑤) = (𝑤 +P 𝑦) | |
6 | 1, 2, 3, 4, 5 | ecovcom 8405 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴)) |
7 | dmaddsr 10509 | . . 3 ⊢ dom +R = (R × R) | |
8 | 7 | ndmovcom 7337 | . 2 ⊢ (¬ (𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴)) |
9 | 6, 8 | pm2.61i 184 | 1 ⊢ (𝐴 +R 𝐵) = (𝐵 +R 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 Pcnp 10283 +P cpp 10285 ~R cer 10288 Rcnr 10289 +R cplr 10293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-ec 8293 df-qs 8297 df-ni 10296 df-pli 10297 df-mi 10298 df-lti 10299 df-plpq 10332 df-mpq 10333 df-ltpq 10334 df-enq 10335 df-nq 10336 df-erq 10337 df-plq 10338 df-mq 10339 df-1nq 10340 df-rq 10341 df-ltnq 10342 df-np 10405 df-plp 10407 df-ltp 10409 df-enr 10479 df-nr 10480 df-plr 10481 |
This theorem is referenced by: pn0sr 10525 sqgt0sr 10530 map2psrpr 10534 axmulcom 10579 axmulass 10581 axdistr 10582 axi2m1 10583 axcnre 10588 |
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