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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 7851. (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 7754 | . 2 ⊢ ℝ = (R × {0R}) | |
2 | oveq1 5843 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (〈𝑥, 𝑦〉 · 1) = (𝐴 · 1)) | |
3 | id 19 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → 〈𝑥, 𝑦〉 = 𝐴) | |
4 | 2, 3 | eqeq12d 2179 | . 2 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → ((〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉 ↔ (𝐴 · 1) = 𝐴)) |
5 | elsni 3588 | . . 3 ⊢ (𝑦 ∈ {0R} → 𝑦 = 0R) | |
6 | df-1 7752 | . . . . . . 7 ⊢ 1 = 〈1R, 0R〉 | |
7 | 6 | oveq2i 5847 | . . . . . 6 ⊢ (〈𝑥, 0R〉 · 1) = (〈𝑥, 0R〉 · 〈1R, 0R〉) |
8 | 1sr 7683 | . . . . . . . 8 ⊢ 1R ∈ R | |
9 | mulresr 7770 | . . . . . . . 8 ⊢ ((𝑥 ∈ R ∧ 1R ∈ R) → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈(𝑥 ·R 1R), 0R〉) | |
10 | 8, 9 | mpan2 422 | . . . . . . 7 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈(𝑥 ·R 1R), 0R〉) |
11 | 1idsr 7700 | . . . . . . . 8 ⊢ (𝑥 ∈ R → (𝑥 ·R 1R) = 𝑥) | |
12 | 11 | opeq1d 3758 | . . . . . . 7 ⊢ (𝑥 ∈ R → 〈(𝑥 ·R 1R), 0R〉 = 〈𝑥, 0R〉) |
13 | 10, 12 | eqtrd 2197 | . . . . . 6 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈𝑥, 0R〉) |
14 | 7, 13 | syl5eq 2209 | . . . . 5 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 1) = 〈𝑥, 0R〉) |
15 | opeq2 3753 | . . . . . . 7 ⊢ (𝑦 = 0R → 〈𝑥, 𝑦〉 = 〈𝑥, 0R〉) | |
16 | 15 | oveq1d 5851 | . . . . . 6 ⊢ (𝑦 = 0R → (〈𝑥, 𝑦〉 · 1) = (〈𝑥, 0R〉 · 1)) |
17 | 16, 15 | eqeq12d 2179 | . . . . 5 ⊢ (𝑦 = 0R → ((〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉 ↔ (〈𝑥, 0R〉 · 1) = 〈𝑥, 0R〉)) |
18 | 14, 17 | syl5ibr 155 | . . . 4 ⊢ (𝑦 = 0R → (𝑥 ∈ R → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉)) |
19 | 18 | impcom 124 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 = 0R) → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉) |
20 | 5, 19 | sylan2 284 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ {0R}) → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉) |
21 | 1, 4, 20 | optocl 4674 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 {csn 3570 〈cop 3573 (class class class)co 5836 Rcnr 7229 0Rc0r 7230 1Rc1r 7231 ·R cmr 7234 ℝcr 7743 1c1 7745 · cmul 7749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-i1p 7399 df-iplp 7400 df-imp 7401 df-enr 7658 df-nr 7659 df-plr 7660 df-mr 7661 df-0r 7663 df-1r 7664 df-m1r 7665 df-c 7750 df-1 7752 df-r 7754 df-mul 7756 |
This theorem is referenced by: rereceu 7821 recriota 7822 |
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