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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 10636 | . . 3 ⊢ (𝜑 → 1 ∈ ℂ) | |
3 | adddirp1d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 1, 2, 3 | adddird 10666 | . 2 ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + (1 · 𝐵))) |
5 | 3 | mulid2d 10659 | . . 3 ⊢ (𝜑 → (1 · 𝐵) = 𝐵) |
6 | 5 | oveq2d 7172 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (1 · 𝐵)) = ((𝐴 · 𝐵) + 𝐵)) |
7 | 4, 6 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 · cmul 10542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-mulcl 10599 ax-mulcom 10601 ax-mulass 10603 ax-distr 10604 ax-1rid 10607 ax-cnre 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: modvalp1 13259 pcexp 16196 mulgnnass 18262 cnfldmulg 20577 dgrcolem1 24863 abelthlem2 25020 2lgsoddprmlem3d 25989 chpdifbndlem1 26129 breprexplemc 31903 fltnltalem 39294 lt3addmuld 41588 lt4addmuld 41593 itgsinexp 42260 fourierdlem19 42431 fourierdlem35 42447 fourierdlem51 42462 |
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