Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ax1rid | Structured version Visualization version GIF version |
Description: 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, derived from ZF set theory. Weakened from the original axiom in the form of statement in mulid1 10638, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 10606. (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 10546 | . 2 ⊢ ℝ = (R × {0R}) | |
2 | oveq1 7162 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (〈𝑥, 𝑦〉 · 1) = (𝐴 · 1)) | |
3 | id 22 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → 〈𝑥, 𝑦〉 = 𝐴) | |
4 | 2, 3 | eqeq12d 2837 | . 2 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → ((〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉 ↔ (𝐴 · 1) = 𝐴)) |
5 | elsni 4583 | . . 3 ⊢ (𝑦 ∈ {0R} → 𝑦 = 0R) | |
6 | df-1 10544 | . . . . . . 7 ⊢ 1 = 〈1R, 0R〉 | |
7 | 6 | oveq2i 7166 | . . . . . 6 ⊢ (〈𝑥, 0R〉 · 1) = (〈𝑥, 0R〉 · 〈1R, 0R〉) |
8 | 1sr 10502 | . . . . . . . 8 ⊢ 1R ∈ R | |
9 | mulresr 10560 | . . . . . . . 8 ⊢ ((𝑥 ∈ R ∧ 1R ∈ R) → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈(𝑥 ·R 1R), 0R〉) | |
10 | 8, 9 | mpan2 689 | . . . . . . 7 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈(𝑥 ·R 1R), 0R〉) |
11 | 1idsr 10519 | . . . . . . . 8 ⊢ (𝑥 ∈ R → (𝑥 ·R 1R) = 𝑥) | |
12 | 11 | opeq1d 4808 | . . . . . . 7 ⊢ (𝑥 ∈ R → 〈(𝑥 ·R 1R), 0R〉 = 〈𝑥, 0R〉) |
13 | 10, 12 | eqtrd 2856 | . . . . . 6 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈𝑥, 0R〉) |
14 | 7, 13 | syl5eq 2868 | . . . . 5 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 1) = 〈𝑥, 0R〉) |
15 | opeq2 4803 | . . . . . . 7 ⊢ (𝑦 = 0R → 〈𝑥, 𝑦〉 = 〈𝑥, 0R〉) | |
16 | 15 | oveq1d 7170 | . . . . . 6 ⊢ (𝑦 = 0R → (〈𝑥, 𝑦〉 · 1) = (〈𝑥, 0R〉 · 1)) |
17 | 16, 15 | eqeq12d 2837 | . . . . 5 ⊢ (𝑦 = 0R → ((〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉 ↔ (〈𝑥, 0R〉 · 1) = 〈𝑥, 0R〉)) |
18 | 14, 17 | syl5ibr 248 | . . . 4 ⊢ (𝑦 = 0R → (𝑥 ∈ R → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉)) |
19 | 18 | impcom 410 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 = 0R) → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉) |
20 | 5, 19 | sylan2 594 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ {0R}) → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉) |
21 | 1, 4, 20 | optocl 5644 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {csn 4566 〈cop 4572 (class class class)co 7155 Rcnr 10286 0Rc0r 10287 1Rc1r 10288 ·R cmr 10291 ℝcr 10535 1c1 10537 · cmul 10541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-omul 8106 df-er 8288 df-ec 8290 df-qs 8294 df-ni 10293 df-pli 10294 df-mi 10295 df-lti 10296 df-plpq 10329 df-mpq 10330 df-ltpq 10331 df-enq 10332 df-nq 10333 df-erq 10334 df-plq 10335 df-mq 10336 df-1nq 10337 df-rq 10338 df-ltnq 10339 df-np 10402 df-1p 10403 df-plp 10404 df-mp 10405 df-ltp 10406 df-enr 10476 df-nr 10477 df-plr 10478 df-mr 10479 df-0r 10481 df-1r 10482 df-m1r 10483 df-c 10542 df-1 10544 df-r 10546 df-mul 10548 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |