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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 11054 | . . 3 ⊢ R = ((P × P) / ~R ) | |
2 | addsrpr 11073 | . . 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 11073 | . . 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 11019 | . . 3 ⊢ (𝑥 +P 𝑧) = (𝑧 +P 𝑥) | |
5 | addcompr 11019 | . . 3 ⊢ (𝑦 +P 𝑤) = (𝑤 +P 𝑦) | |
6 | 1, 2, 3, 4, 5 | ecovcom 8820 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴)) |
7 | dmaddsr 11083 | . . 3 ⊢ dom +R = (R × R) | |
8 | 7 | ndmovcom 7597 | . 2 ⊢ (¬ (𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴)) |
9 | 6, 8 | pm2.61i 182 | 1 ⊢ (𝐴 +R 𝐵) = (𝐵 +R 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2105 (class class class)co 7412 Pcnp 10857 +P cpp 10859 ~R cer 10862 Rcnr 10863 +R cplr 10867 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-inf2 9639 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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-int 4951 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 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-oadd 8473 df-omul 8474 df-er 8706 df-ec 8708 df-qs 8712 df-ni 10870 df-pli 10871 df-mi 10872 df-lti 10873 df-plpq 10906 df-mpq 10907 df-ltpq 10908 df-enq 10909 df-nq 10910 df-erq 10911 df-plq 10912 df-mq 10913 df-1nq 10914 df-rq 10915 df-ltnq 10916 df-np 10979 df-plp 10981 df-ltp 10983 df-enr 11053 df-nr 11054 df-plr 11055 |
This theorem is referenced by: pn0sr 11099 sqgt0sr 11104 map2psrpr 11108 axmulcom 11153 axmulass 11155 axdistr 11156 axi2m1 11157 axcnre 11162 |
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