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Mirrors > Home > MPE Home > Th. List > Mathboxes > int-mul11d | Structured version Visualization version GIF version |
Description: First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
Ref | Expression |
---|---|
int-mul11d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
int-mul11d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
int-mul11d | ⊢ (𝜑 → (𝐴 · 1) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | int-mul11d.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 10668 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 2 | mulid1d 10657 | . 2 ⊢ (𝜑 → (𝐴 · 1) = 𝐴) |
4 | int-mul11d.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
5 | 3, 4 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝐴 · 1) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 (class class class)co 7155 ℝcr 10535 1c1 10537 · cmul 10541 |
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-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-mulcl 10598 ax-mulcom 10600 ax-mulass 10602 ax-distr 10603 ax-1rid 10606 ax-cnre 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 |
This theorem is referenced by: (None) |
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