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Theorem int-mul12d 37508
 Description: Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
Hypotheses
Ref Expression
int-mul12d.1 (𝜑𝐴 ∈ ℝ)
int-mul12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
int-mul12d (𝜑 → (1 · 𝐴) = 𝐵)

Proof of Theorem int-mul12d
StepHypRef Expression
1 int-mul12d.1 . . . 4 (𝜑𝐴 ∈ ℝ)
21recnd 9947 . . 3 (𝜑𝐴 ∈ ℂ)
32mulid2d 9937 . 2 (𝜑 → (1 · 𝐴) = 𝐴)
4 int-mul12d.2 . 2 (𝜑𝐴 = 𝐵)
53, 4eqtrd 2644 1 (𝜑 → (1 · 𝐴) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  (class class class)co 6549  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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-mulcom 9879  ax-mulass 9881  ax-distr 9882  ax-1rid 9885  ax-cnre 9888 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by: (None)
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