Mathbox for Stanislas Polu |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > int-mul12d | Structured version Visualization version GIF version |
Description: Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
Ref | Expression |
---|---|
int-mul12d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
int-mul12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
int-mul12d | ⊢ (𝜑 → (1 · 𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | int-mul12d.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 10661 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 2 | mulid2d 10651 | . 2 ⊢ (𝜑 → (1 · 𝐴) = 𝐴) |
4 | int-mul12d.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
5 | 3, 4 | eqtrd 2854 | 1 ⊢ (𝜑 → (1 · 𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 (class class class)co 7148 ℝcr 10528 1c1 10530 · cmul 10534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-mulcl 10591 ax-mulcom 10593 ax-mulass 10595 ax-distr 10596 ax-1rid 10599 ax-cnre 10602 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7151 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |