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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 9916, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 9885. (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 9825 | . 2 ⊢ ℝ = (R × {0R}) | |
2 | oveq1 6556 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (〈𝑥, 𝑦〉 · 1) = (𝐴 · 1)) | |
3 | id 22 | . . 3 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → 〈𝑥, 𝑦〉 = 𝐴) | |
4 | 2, 3 | eqeq12d 2625 | . 2 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → ((〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉 ↔ (𝐴 · 1) = 𝐴)) |
5 | elsni 4142 | . . 3 ⊢ (𝑦 ∈ {0R} → 𝑦 = 0R) | |
6 | df-1 9823 | . . . . . . 7 ⊢ 1 = 〈1R, 0R〉 | |
7 | 6 | oveq2i 6560 | . . . . . 6 ⊢ (〈𝑥, 0R〉 · 1) = (〈𝑥, 0R〉 · 〈1R, 0R〉) |
8 | 1sr 9781 | . . . . . . . 8 ⊢ 1R ∈ R | |
9 | mulresr 9839 | . . . . . . . 8 ⊢ ((𝑥 ∈ R ∧ 1R ∈ R) → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈(𝑥 ·R 1R), 0R〉) | |
10 | 8, 9 | mpan2 703 | . . . . . . 7 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈(𝑥 ·R 1R), 0R〉) |
11 | 1idsr 9798 | . . . . . . . 8 ⊢ (𝑥 ∈ R → (𝑥 ·R 1R) = 𝑥) | |
12 | 11 | opeq1d 4346 | . . . . . . 7 ⊢ (𝑥 ∈ R → 〈(𝑥 ·R 1R), 0R〉 = 〈𝑥, 0R〉) |
13 | 10, 12 | eqtrd 2644 | . . . . . 6 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 〈1R, 0R〉) = 〈𝑥, 0R〉) |
14 | 7, 13 | syl5eq 2656 | . . . . 5 ⊢ (𝑥 ∈ R → (〈𝑥, 0R〉 · 1) = 〈𝑥, 0R〉) |
15 | opeq2 4341 | . . . . . . 7 ⊢ (𝑦 = 0R → 〈𝑥, 𝑦〉 = 〈𝑥, 0R〉) | |
16 | 15 | oveq1d 6564 | . . . . . 6 ⊢ (𝑦 = 0R → (〈𝑥, 𝑦〉 · 1) = (〈𝑥, 0R〉 · 1)) |
17 | 16, 15 | eqeq12d 2625 | . . . . 5 ⊢ (𝑦 = 0R → ((〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉 ↔ (〈𝑥, 0R〉 · 1) = 〈𝑥, 0R〉)) |
18 | 14, 17 | syl5ibr 235 | . . . 4 ⊢ (𝑦 = 0R → (𝑥 ∈ R → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉)) |
19 | 18 | impcom 445 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 = 0R) → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉) |
20 | 5, 19 | sylan2 490 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ {0R}) → (〈𝑥, 𝑦〉 · 1) = 〈𝑥, 𝑦〉) |
21 | 1, 4, 20 | optocl 5118 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {csn 4125 〈cop 4131 (class class class)co 6549 Rcnr 9566 0Rc0r 9567 1Rc1r 9568 ·R cmr 9571 ℝcr 9814 1c1 9816 · cmul 9820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-ni 9573 df-pli 9574 df-mi 9575 df-lti 9576 df-plpq 9609 df-mpq 9610 df-ltpq 9611 df-enq 9612 df-nq 9613 df-erq 9614 df-plq 9615 df-mq 9616 df-1nq 9617 df-rq 9618 df-ltnq 9619 df-np 9682 df-1p 9683 df-plp 9684 df-mp 9685 df-ltp 9686 df-enr 9756 df-nr 9757 df-plr 9758 df-mr 9759 df-0r 9761 df-1r 9762 df-m1r 9763 df-c 9821 df-1 9823 df-r 9825 df-mul 9827 |
This theorem is referenced by: (None) |
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