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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 10633, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 10601. (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 10541 | . 2 ⊢ ℝ = (R × {0R}) | |
2 | oveq1 7157 | . . 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 4577 | . . 3 ⊢ (𝑦 ∈ {0R} → 𝑦 = 0R) | |
6 | df-1 10539 | . . . . . . 7 ⊢ 1 = 〈1R, 0R〉 | |
7 | 6 | oveq2i 7161 | . . . . . 6 ⊢ (〈𝑥, 0R〉 · 1) = (〈𝑥, 0R〉 · 〈1R, 0R〉) |
8 | 1sr 10497 | . . . . . . . 8 ⊢ 1R ∈ R | |
9 | mulresr 10555 | . . . . . . . 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 10514 | . . . . . . . 8 ⊢ (𝑥 ∈ R → (𝑥 ·R 1R) = 𝑥) | |
12 | 11 | opeq1d 4802 | . . . . . . 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 4797 | . . . . . . 7 ⊢ (𝑦 = 0R → 〈𝑥, 𝑦〉 = 〈𝑥, 0R〉) | |
16 | 15 | oveq1d 7165 | . . . . . 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 5639 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {csn 4560 〈cop 4566 (class class class)co 7150 Rcnr 10281 0Rc0r 10282 1Rc1r 10283 ·R cmr 10286 ℝcr 10530 1c1 10532 · cmul 10536 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-rq 10333 df-ltnq 10334 df-np 10397 df-1p 10398 df-plp 10399 df-mp 10400 df-ltp 10401 df-enr 10471 df-nr 10472 df-plr 10473 df-mr 10474 df-0r 10476 df-1r 10477 df-m1r 10478 df-c 10537 df-1 10539 df-r 10541 df-mul 10543 |
This theorem is referenced by: (None) |
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