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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 7740 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | addsrpr 7758 | . 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 7758 | . 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 7591 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑧 ∈ P) → (𝑥 +P 𝑧) = (𝑧 +P 𝑥)) | |
5 | 4 | ad2ant2r 509 | . 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑥 +P 𝑧) = (𝑧 +P 𝑥)) |
6 | addcomprg 7591 | . . 3 ⊢ ((𝑦 ∈ P ∧ 𝑤 ∈ P) → (𝑦 +P 𝑤) = (𝑤 +P 𝑦)) | |
7 | 6 | ad2ant2l 508 | . 2 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (𝑧 ∈ P ∧ 𝑤 ∈ P)) → (𝑦 +P 𝑤) = (𝑤 +P 𝑦)) |
8 | 1, 2, 3, 5, 7 | ecovicom 6657 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) = (𝐵 +R 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 (class class class)co 5888 Pcnp 7304 +P cpp 7306 ~R cer 7309 Rcnr 7310 +R cplr 7314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-eprel 4301 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-recs 6320 df-irdg 6385 df-1o 6431 df-2o 6432 df-oadd 6435 df-omul 6436 df-er 6549 df-ec 6551 df-qs 6555 df-ni 7317 df-pli 7318 df-mi 7319 df-lti 7320 df-plpq 7357 df-mpq 7358 df-enq 7360 df-nqqs 7361 df-plqqs 7362 df-mqqs 7363 df-1nqqs 7364 df-rq 7365 df-ltnqqs 7366 df-enq0 7437 df-nq0 7438 df-0nq0 7439 df-plq0 7440 df-mq0 7441 df-inp 7479 df-iplp 7481 df-enr 7739 df-nr 7740 df-plr 7741 |
This theorem is referenced by: pn0sr 7784 caucvgsrlemoffval 7809 caucvgsrlemoffcau 7811 caucvgsrlemoffgt1 7812 caucvgsrlemoffres 7813 caucvgsr 7815 map2psrprg 7818 axaddcom 7883 axmulcom 7884 axmulass 7886 axdistr 7887 axi2m1 7888 axcnre 7894 |
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