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| 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 11256 | . . 3 ⊢ (𝜑 → 1 ∈ ℂ) | |
| 3 | adddirp1d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 4 | 1, 2, 3 | adddird 11286 | . 2 ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + (1 · 𝐵))) |
| 5 | 3 | mullidd 11279 | . . 3 ⊢ (𝜑 → (1 · 𝐵) = 𝐵) |
| 6 | 5 | oveq2d 7447 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (1 · 𝐵)) = ((𝐴 · 𝐵) + 𝐵)) |
| 7 | 4, 6 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-mulcom 11219 ax-mulass 11221 ax-distr 11222 ax-1rid 11225 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: modvalp1 13930 pcexp 16897 mulgnnass 19127 cnfldmulg 21416 dgrcolem1 26313 abelthlem2 26476 2lgsoddprmlem3d 27457 chpdifbndlem1 27597 breprexplemc 34647 deg1pow 42142 fltnltalem 42672 lt3addmuld 45313 lt4addmuld 45318 itgsinexp 45970 fourierdlem19 46141 fourierdlem35 46157 fourierdlem51 46172 minusmodnep2tmod 47355 gpg3kgrtriexlem2 48040 |
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