Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > adddirp1d | Structured version Visualization version GIF version |
Description: Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
adddirp1d.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
adddirp1d.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
adddirp1d | ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | adddirp1d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | 1cnd 10056 | . . 3 ⊢ (𝜑 → 1 ∈ ℂ) | |
3 | adddirp1d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 1, 2, 3 | adddird 10065 | . 2 ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + (1 · 𝐵))) |
5 | 3 | mulid2d 10058 | . . 3 ⊢ (𝜑 → (1 · 𝐵) = 𝐵) |
6 | 5 | oveq2d 6666 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (1 · 𝐵)) = ((𝐴 · 𝐵) + 𝐵)) |
7 | 4, 6 | eqtrd 2656 | 1 ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-mulcom 10000 ax-mulass 10002 ax-distr 10003 ax-1rid 10006 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: modvalp1 12689 2lgsoddprmlem3d 25138 breprexplemc 30710 lt3addmuld 39515 lt4addmuld 39520 itgsinexp 40170 fourierdlem19 40343 fourierdlem35 40359 fourierdlem51 40374 |
Copyright terms: Public domain | W3C validator |