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Theorem addcomsr 9905
Description: Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
addcomsr (𝐴 +R 𝐵) = (𝐵 +R 𝐴)

Proof of Theorem addcomsr
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 9875 . . 3 R = ((P × P) / ~R )
2 addsrpr 9893 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R +R [⟨𝑧, 𝑤⟩] ~R ) = [⟨(𝑥 +P 𝑧), (𝑦 +P 𝑤)⟩] ~R )
3 addsrpr 9893 . . 3 (((𝑧P𝑤P) ∧ (𝑥P𝑦P)) → ([⟨𝑧, 𝑤⟩] ~R +R [⟨𝑥, 𝑦⟩] ~R ) = [⟨(𝑧 +P 𝑥), (𝑤 +P 𝑦)⟩] ~R )
4 addcompr 9840 . . 3 (𝑥 +P 𝑧) = (𝑧 +P 𝑥)
5 addcompr 9840 . . 3 (𝑦 +P 𝑤) = (𝑤 +P 𝑦)
61, 2, 3, 4, 5ecovcom 7851 . 2 ((𝐴R𝐵R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴))
7 dmaddsr 9903 . . 3 dom +R = (R × R)
87ndmovcom 6818 . 2 (¬ (𝐴R𝐵R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴))
96, 8pm2.61i 176 1 (𝐴 +R 𝐵) = (𝐵 +R 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1482  wcel 1989  (class class class)co 6647  Pcnp 9678   +P cpp 9680   ~R cer 9683  Rcnr 9684   +R cplr 9688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-omul 7562  df-er 7739  df-ec 7741  df-qs 7745  df-ni 9691  df-pli 9692  df-mi 9693  df-lti 9694  df-plpq 9727  df-mpq 9728  df-ltpq 9729  df-enq 9730  df-nq 9731  df-erq 9732  df-plq 9733  df-mq 9734  df-1nq 9735  df-rq 9736  df-ltnq 9737  df-np 9800  df-plp 9802  df-ltp 9804  df-enr 9874  df-nr 9875  df-plr 9876
This theorem is referenced by:  pn0sr  9919  sqgt0sr  9924  map2psrpr  9928  axmulcom  9973  axmulass  9975  axdistr  9976  axi2m1  9977  axcnre  9982
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