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Theorem ax1rid 10184
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 10239, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 10208. (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.
StepHypRef Expression
1 df-r 10148 . 2 ℝ = (R × {0R})
2 oveq1 6800 . . 3 (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ · 1) = (𝐴 · 1))
3 id 22 . . 3 (⟨𝑥, 𝑦⟩ = 𝐴 → ⟨𝑥, 𝑦⟩ = 𝐴)
42, 3eqeq12d 2786 . 2 (⟨𝑥, 𝑦⟩ = 𝐴 → ((⟨𝑥, 𝑦⟩ · 1) = ⟨𝑥, 𝑦⟩ ↔ (𝐴 · 1) = 𝐴))
5 elsni 4333 . . 3 (𝑦 ∈ {0R} → 𝑦 = 0R)
6 df-1 10146 . . . . . . 7 1 = ⟨1R, 0R
76oveq2i 6804 . . . . . 6 (⟨𝑥, 0R⟩ · 1) = (⟨𝑥, 0R⟩ · ⟨1R, 0R⟩)
8 1sr 10104 . . . . . . . 8 1RR
9 mulresr 10162 . . . . . . . 8 ((𝑥R ∧ 1RR) → (⟨𝑥, 0R⟩ · ⟨1R, 0R⟩) = ⟨(𝑥 ·R 1R), 0R⟩)
108, 9mpan2 671 . . . . . . 7 (𝑥R → (⟨𝑥, 0R⟩ · ⟨1R, 0R⟩) = ⟨(𝑥 ·R 1R), 0R⟩)
11 1idsr 10121 . . . . . . . 8 (𝑥R → (𝑥 ·R 1R) = 𝑥)
1211opeq1d 4545 . . . . . . 7 (𝑥R → ⟨(𝑥 ·R 1R), 0R⟩ = ⟨𝑥, 0R⟩)
1310, 12eqtrd 2805 . . . . . 6 (𝑥R → (⟨𝑥, 0R⟩ · ⟨1R, 0R⟩) = ⟨𝑥, 0R⟩)
147, 13syl5eq 2817 . . . . 5 (𝑥R → (⟨𝑥, 0R⟩ · 1) = ⟨𝑥, 0R⟩)
15 opeq2 4540 . . . . . . 7 (𝑦 = 0R → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 0R⟩)
1615oveq1d 6808 . . . . . 6 (𝑦 = 0R → (⟨𝑥, 𝑦⟩ · 1) = (⟨𝑥, 0R⟩ · 1))
1716, 15eqeq12d 2786 . . . . 5 (𝑦 = 0R → ((⟨𝑥, 𝑦⟩ · 1) = ⟨𝑥, 𝑦⟩ ↔ (⟨𝑥, 0R⟩ · 1) = ⟨𝑥, 0R⟩))
1814, 17syl5ibr 236 . . . 4 (𝑦 = 0R → (𝑥R → (⟨𝑥, 𝑦⟩ · 1) = ⟨𝑥, 𝑦⟩))
1918impcom 394 . . 3 ((𝑥R𝑦 = 0R) → (⟨𝑥, 𝑦⟩ · 1) = ⟨𝑥, 𝑦⟩)
205, 19sylan2 580 . 2 ((𝑥R𝑦 ∈ {0R}) → (⟨𝑥, 𝑦⟩ · 1) = ⟨𝑥, 𝑦⟩)
211, 4, 20optocl 5335 1 (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {csn 4316  cop 4322  (class class class)co 6793  Rcnr 9889  0Rc0r 9890  1Rc1r 9891   ·R cmr 9894  cr 10137  1c1 10139   · cmul 10143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-er 7896  df-ec 7898  df-qs 7902  df-ni 9896  df-pli 9897  df-mi 9898  df-lti 9899  df-plpq 9932  df-mpq 9933  df-ltpq 9934  df-enq 9935  df-nq 9936  df-erq 9937  df-plq 9938  df-mq 9939  df-1nq 9940  df-rq 9941  df-ltnq 9942  df-np 10005  df-1p 10006  df-plp 10007  df-mp 10008  df-ltp 10009  df-enr 10079  df-nr 10080  df-plr 10081  df-mr 10082  df-0r 10084  df-1r 10085  df-m1r 10086  df-c 10144  df-1 10146  df-r 10148  df-mul 10150
This theorem is referenced by: (None)
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