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Mirrors > Home > ILE Home > Th. List > ax1rid | GIF version |
Description: 1 is an identity element for real multiplication. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 7932. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax1rid | ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-r 7835 | . 2 ⊢ ℝ = (R × {0R}) | |
2 | oveq1 5895 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (〈𝑥, 𝑦〉 · 1) = (𝐴 · 1)) | |
3 | id 19 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → 〈𝑥, 𝑦〉 = 𝐴) | |
4 | 2, 3 | eqeq12d 2202 | . 2 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → ((〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉 ↔ (𝐴 · 1) = 𝐴)) |
5 | elsni 3622 | . . 3 ⊢ (𝑦 ∈ {0R} → 𝑦 = 0R) | |
6 | df-1 7833 | . . . . . . 7 ⊢ 1 = 〈1R, 0R〉 | |
7 | 6 | oveq2i 5899 | . . . . . 6 ⊢ (〈𝑥, 0R〉 · 1) = (〈𝑥, 0R〉 · 〈1R, 0R〉) |
8 | 1sr 7764 | . . . . . . . 8 ⊢ 1R ∈ R | |
9 | mulresr 7851 | . . . . . . . 8 ⊢ ((𝑥 ∈ R ∧ 1R ∈ R) → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈(𝑥 ·R 1R), 0R〉) | |
10 | 8, 9 | mpan2 425 | . . . . . . 7 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈(𝑥 ·R 1R), 0R〉) |
11 | 1idsr 7781 | . . . . . . . 8 ⊢ (𝑥 ∈ R → (𝑥 ·R 1R) = 𝑥) | |
12 | 11 | opeq1d 3796 | . . . . . . 7 ⊢ (𝑥 ∈ R → 〈(𝑥 ·R 1R), 0R〉 = 〈𝑥, 0R〉) |
13 | 10, 12 | eqtrd 2220 | . . . . . 6 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈𝑥, 0R〉) |
14 | 7, 13 | eqtrid 2232 | . . . . 5 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 1) = 〈𝑥, 0R〉) |
15 | opeq2 3791 | . . . . . . 7 ⊢ (𝑦 = 0R → 〈𝑥, 𝑦〉 = 〈𝑥, 0R〉) | |
16 | 15 | oveq1d 5903 | . . . . . 6 ⊢ (𝑦 = 0R → (〈𝑥, 𝑦〉 · 1) = (〈𝑥, 0R〉 · 1)) |
17 | 16, 15 | eqeq12d 2202 | . . . . 5 ⊢ (𝑦 = 0R → ((〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉 ↔ (〈𝑥, 0R〉 · 1) = 〈𝑥, 0R〉)) |
18 | 14, 17 | imbitrrid 156 | . . . 4 ⊢ (𝑦 = 0R → (𝑥 ∈ R → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉)) |
19 | 18 | impcom 125 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 = 0R) → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉) |
20 | 5, 19 | sylan2 286 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ {0R}) → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉) |
21 | 1, 4, 20 | optocl 4714 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 {csn 3604 〈cop 3607 (class class class)co 5888 Rcnr 7310 0Rc0r 7311 1Rc1r 7312 ·R cmr 7315 ℝcr 7824 1c1 7826 · cmul 7830 |
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-i1p 7480 df-iplp 7481 df-imp 7482 df-enr 7739 df-nr 7740 df-plr 7741 df-mr 7742 df-0r 7744 df-1r 7745 df-m1r 7746 df-c 7831 df-1 7833 df-r 7835 df-mul 7837 |
This theorem is referenced by: rereceu 7902 recriota 7903 |
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