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Mirrors > Home > MPE Home > Th. List > 0idsr | Structured version Visualization version GIF version |
Description: The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0idsr | ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 10466 | . 2 ⊢ R = ((P × P) / ~R ) | |
2 | oveq1 7152 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → ([〈𝑥, 𝑦〉] ~R +R 0R) = (𝐴 +R 0R)) | |
3 | id 22 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → [〈𝑥, 𝑦〉] ~R = 𝐴) | |
4 | 2, 3 | eqeq12d 2834 | . 2 ⊢ ([〈𝑥, 𝑦〉] ~R = 𝐴 → (([〈𝑥, 𝑦〉] ~R +R 0R) = [〈𝑥, 𝑦〉] ~R ↔ (𝐴 +R 0R) = 𝐴)) |
5 | df-0r 10470 | . . . 4 ⊢ 0R = [〈1P, 1P〉] ~R | |
6 | 5 | oveq2i 7156 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~R +R 0R) = ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) |
7 | 1pr 10425 | . . . . 5 ⊢ 1P ∈ P | |
8 | addsrpr 10485 | . . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ (1P ∈ P ∧ 1P ∈ P)) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) | |
9 | 7, 7, 8 | mpanr12 701 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) |
10 | addclpr 10428 | . . . . . . 7 ⊢ ((𝑥 ∈ P ∧ 1P ∈ P) → (𝑥 +P 1P) ∈ P) | |
11 | 7, 10 | mpan2 687 | . . . . . 6 ⊢ (𝑥 ∈ P → (𝑥 +P 1P) ∈ P) |
12 | addclpr 10428 | . . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 +P 1P) ∈ P) | |
13 | 7, 12 | mpan2 687 | . . . . . 6 ⊢ (𝑦 ∈ P → (𝑦 +P 1P) ∈ P) |
14 | 11, 13 | anim12i 612 | . . . . 5 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) |
15 | vex 3495 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
16 | vex 3495 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
17 | 7 | elexi 3511 | . . . . . . 7 ⊢ 1P ∈ V |
18 | addcompr 10431 | . . . . . . 7 ⊢ (𝑧 +P 𝑤) = (𝑤 +P 𝑧) | |
19 | addasspr 10432 | . . . . . . 7 ⊢ ((𝑧 +P 𝑤) +P 𝑣) = (𝑧 +P (𝑤 +P 𝑣)) | |
20 | 15, 16, 17, 18, 19 | caov12 7365 | . . . . . 6 ⊢ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P)) |
21 | enreceq 10476 | . . . . . 6 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) → ([〈𝑥, 𝑦〉] ~R = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ↔ (𝑥 +P (𝑦 +P 1P)) = (𝑦 +P (𝑥 +P 1P)))) | |
22 | 20, 21 | mpbiri 259 | . . . . 5 ⊢ (((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧ ((𝑥 +P 1P) ∈ P ∧ (𝑦 +P 1P) ∈ P)) → [〈𝑥, 𝑦〉] ~R = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) |
23 | 14, 22 | mpdan 683 | . . . 4 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → [〈𝑥, 𝑦〉] ~R = [〈(𝑥 +P 1P), (𝑦 +P 1P)〉] ~R ) |
24 | 9, 23 | eqtr4d 2856 | . . 3 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R [〈1P, 1P〉] ~R ) = [〈𝑥, 𝑦〉] ~R ) |
25 | 6, 24 | syl5eq 2865 | . 2 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → ([〈𝑥, 𝑦〉] ~R +R 0R) = [〈𝑥, 𝑦〉] ~R ) |
26 | 1, 4, 25 | ecoptocl 8376 | 1 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 〈cop 4563 (class class class)co 7145 [cec 8276 Pcnp 10269 1Pc1p 10270 +P cpp 10271 ~R cer 10274 Rcnr 10275 0Rc0r 10276 +R cplr 10279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-omul 8096 df-er 8278 df-ec 8280 df-qs 8284 df-ni 10282 df-pli 10283 df-mi 10284 df-lti 10285 df-plpq 10318 df-mpq 10319 df-ltpq 10320 df-enq 10321 df-nq 10322 df-erq 10323 df-plq 10324 df-mq 10325 df-1nq 10326 df-rq 10327 df-ltnq 10328 df-np 10391 df-1p 10392 df-plp 10393 df-ltp 10395 df-enr 10465 df-nr 10466 df-plr 10467 df-0r 10470 |
This theorem is referenced by: addgt0sr 10514 sqgt0sr 10516 map2psrpr 10520 supsrlem 10521 addresr 10548 mulresr 10549 axi2m1 10569 axcnre 10574 |
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