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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 10672 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | 2 | mulid1d 10661 | . 2 ⊢ (𝜑 → (𝐴 · 1) = 𝐴) |
4 | int-mul11d.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
5 | 3, 4 | eqtrd 2859 | 1 ⊢ (𝜑 → (𝐴 · 1) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 (class class class)co 7159 ℝcr 10539 1c1 10541 · cmul 10545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-mulcl 10602 ax-mulcom 10604 ax-mulass 10606 ax-distr 10607 ax-1rid 10610 ax-cnre 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 |
This theorem is referenced by: (None) |
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