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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 9757 | . . 3 ⊢ R = ((P × P) / ~R ) | |
2 | addsrpr 9775 | . . 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 9775 | . . 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 9722 | . . 3 ⊢ (𝑥 +P 𝑧) = (𝑧 +P 𝑥) | |
5 | addcompr 9722 | . . 3 ⊢ (𝑦 +P 𝑤) = (𝑤 +P 𝑦) | |
6 | 1, 2, 3, 4, 5 | ecovcom 7741 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴)) |
7 | dmaddsr 9785 | . . 3 ⊢ dom +R = (R × R) | |
8 | 7 | ndmovcom 6719 | . 2 ⊢ (¬ (𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴)) |
9 | 6, 8 | pm2.61i 175 | 1 ⊢ (𝐴 +R 𝐵) = (𝐵 +R 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 Pcnp 9560 +P cpp 9562 ~R cer 9565 Rcnr 9566 +R cplr 9570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-ni 9573 df-pli 9574 df-mi 9575 df-lti 9576 df-plpq 9609 df-mpq 9610 df-ltpq 9611 df-enq 9612 df-nq 9613 df-erq 9614 df-plq 9615 df-mq 9616 df-1nq 9617 df-rq 9618 df-ltnq 9619 df-np 9682 df-plp 9684 df-ltp 9686 df-enr 9756 df-nr 9757 df-plr 9758 |
This theorem is referenced by: pn0sr 9801 sqgt0sr 9806 map2psrpr 9810 axmulcom 9855 axmulass 9857 axdistr 9858 axi2m1 9859 axcnre 9864 |
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