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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 11254 | . . 3 ⊢ (𝜑 → 1 ∈ ℂ) | |
3 | adddirp1d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 1, 2, 3 | adddird 11284 | . 2 ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + (1 · 𝐵))) |
5 | 3 | mullidd 11277 | . . 3 ⊢ (𝜑 → (1 · 𝐵) = 𝐵) |
6 | 5 | oveq2d 7447 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (1 · 𝐵)) = ((𝐴 · 𝐵) + 𝐵)) |
7 | 4, 6 | eqtrd 2775 | 1 ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 1c1 11154 + caddc 11156 · cmul 11158 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-mulcom 11217 ax-mulass 11219 ax-distr 11220 ax-1rid 11223 ax-cnre 11226 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: modvalp1 13927 pcexp 16893 mulgnnass 19140 cnfldmulg 21434 dgrcolem1 26328 abelthlem2 26491 2lgsoddprmlem3d 27472 chpdifbndlem1 27612 breprexplemc 34626 deg1pow 42123 fltnltalem 42649 lt3addmuld 45252 lt4addmuld 45257 itgsinexp 45911 fourierdlem19 46082 fourierdlem35 46098 fourierdlem51 46113 minusmodnep2tmod 47293 |
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