Theorem ax1rid 11059

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 mulrid 11117, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 11083. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ax1rid	⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)

Proof of Theorem ax1rid

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-r 11023	. 2 ⊢ ℝ = (R × {0R})
2		oveq1 7359	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ · 1) = (𝐴 · 1))
3		id 22	. . 3 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ⟨𝑥, 𝑦⟩ = 𝐴)
4	2, 3	eqeq12d 2749	. 2 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ · 1) = ⟨𝑥, 𝑦⟩ ↔ (𝐴 · 1) = 𝐴))
5		elsni 4592	. . 3 ⊢ (𝑦 ∈ {0R} → 𝑦 = 0R)
6		df-1 11021	. . . . . . 7 ⊢ 1 = ⟨1R, 0R⟩
7	6	oveq2i 7363	. . . . . 6 ⊢ (⟨𝑥, 0R⟩ · 1) = (⟨𝑥, 0R⟩ · ⟨1R, 0R⟩)
8		1sr 10979	. . . . . . . 8 ⊢ 1R ∈ R
9		mulresr 11037	. . . . . . . 8 ⊢ ((𝑥 ∈ R ∧ 1R ∈ R) → (⟨𝑥, 0R⟩ · ⟨1R, 0R⟩) = ⟨(𝑥 ·R 1R), 0R⟩)
10	8, 9	mpan2 691	. . . . . . 7 ⊢ (𝑥 ∈ R → (⟨𝑥, 0R⟩ · ⟨1R, 0R⟩) = ⟨(𝑥 ·R 1R), 0R⟩)
11		1idsr 10996	. . . . . . . 8 ⊢ (𝑥 ∈ R → (𝑥 ·R 1R) = 𝑥)
12	11	opeq1d 4830	. . . . . . 7 ⊢ (𝑥 ∈ R → ⟨(𝑥 ·R 1R), 0R⟩ = ⟨𝑥, 0R⟩)
13	10, 12	eqtrd 2768	. . . . . 6 ⊢ (𝑥 ∈ R → (⟨𝑥, 0R⟩ · ⟨1R, 0R⟩) = ⟨𝑥, 0R⟩)
14	7, 13	eqtrid 2780	. . . . 5 ⊢ (𝑥 ∈ R → (⟨𝑥, 0R⟩ · 1) = ⟨𝑥, 0R⟩)
15		opeq2 4825	. . . . . . 7 ⊢ (𝑦 = 0R → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 0R⟩)
16	15	oveq1d 7367	. . . . . 6 ⊢ (𝑦 = 0R → (⟨𝑥, 𝑦⟩ · 1) = (⟨𝑥, 0R⟩ · 1))
17	16, 15	eqeq12d 2749	. . . . 5 ⊢ (𝑦 = 0R → ((⟨𝑥, 𝑦⟩ · 1) = ⟨𝑥, 𝑦⟩ ↔ (⟨𝑥, 0R⟩ · 1) = ⟨𝑥, 0R⟩))
18	14, 17	imbitrrid 246	. . . 4 ⊢ (𝑦 = 0R → (𝑥 ∈ R → (⟨𝑥, 𝑦⟩ · 1) = ⟨𝑥, 𝑦⟩))
19	18	impcom 407	. . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 = 0R) → (⟨𝑥, 𝑦⟩ · 1) = ⟨𝑥, 𝑦⟩)
20	5, 19	sylan2 593	. 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ {0R}) → (⟨𝑥, 𝑦⟩ · 1) = ⟨𝑥, 𝑦⟩)
21	1, 4, 20	optocl 5713	1 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {csn 4575  ⟨cop 4581  (class class class)co 7352  Rcnr 10763  0_Rc0r 10764  1_Rc1r 10765   ·_R cmr 10768  ℝcr 11012  1c1 11014   · cmul 11018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-oadd 8395  df-omul 8396  df-er 8628  df-ec 8630  df-qs 8634  df-ni 10770  df-pli 10771  df-mi 10772  df-lti 10773  df-plpq 10806  df-mpq 10807  df-ltpq 10808  df-enq 10809  df-nq 10810  df-erq 10811  df-plq 10812  df-mq 10813  df-1nq 10814  df-rq 10815  df-ltnq 10816  df-np 10879  df-1p 10880  df-plp 10881  df-mp 10882  df-ltp 10883  df-enr 10953  df-nr 10954  df-plr 10955  df-mr 10956  df-0r 10958  df-1r 10959  df-m1r 10960  df-c 11019  df-1 11021  df-r 11023  df-mul 11025

This theorem is referenced by: (None)