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Mirrors > Home > ILE Home > Th. List > addcomsrg | GIF version |
Description: Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Ref | Expression |
---|---|
addcomsrg | ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 7174 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | addsrpr 7192 | . 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → ([〈𝑥, 𝑦〉] ~R +R [〈𝑧, 𝑤〉] ~R ) = [〈(𝑥 +P 𝑧), (𝑦 +P 𝑤)〉] ~R ) | |
3 | addsrpr 7192 | . 2 ⊢ (((𝑧 ∈ P ∧ 𝑤 ∈ P) ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → ([〈𝑧, 𝑤〉] ~R +R [〈𝑥, 𝑦〉] ~R ) = [〈(𝑧 +P 𝑥), (𝑤 +P 𝑦)〉] ~R ) | |
4 | addcomprg 7038 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 +P 𝑧) = (𝑧 +P 𝑥)) | |
5 | 4 | ad2ant2r 493 | . 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑥 +P 𝑧) = (𝑧 +P 𝑥)) |
6 | addcomprg 7038 | . . 3 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 +P 𝑤) = (𝑤 +P 𝑦)) | |
7 | 6 | ad2ant2l 492 | . 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦 +P 𝑤) = (𝑤 +P 𝑦)) |
8 | 1, 2, 3, 5, 7 | ecovicom 6328 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 (class class class)co 5589 Pcnp 6751 +P cpp 6753 ~R cer 6756 Rcnr 6757 +R cplr 6761 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-eprel 4079 df-id 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-recs 6000 df-irdg 6065 df-1o 6111 df-2o 6112 df-oadd 6115 df-omul 6116 df-er 6220 df-ec 6222 df-qs 6226 df-ni 6764 df-pli 6765 df-mi 6766 df-lti 6767 df-plpq 6804 df-mpq 6805 df-enq 6807 df-nqqs 6808 df-plqqs 6809 df-mqqs 6810 df-1nqqs 6811 df-rq 6812 df-ltnqqs 6813 df-enq0 6884 df-nq0 6885 df-0nq0 6886 df-plq0 6887 df-mq0 6888 df-inp 6926 df-iplp 6928 df-enr 7173 df-nr 7174 df-plr 7175 |
This theorem is referenced by: pn0sr 7218 caucvgsrlemoffval 7242 caucvgsrlemoffcau 7244 caucvgsrlemoffgt1 7245 caucvgsrlemoffres 7246 caucvgsr 7248 axaddcom 7306 axmulcom 7307 axmulass 7309 axdistr 7310 axi2m1 7311 axcnre 7317 |
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