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Mirrors > Home > ILE Home > Th. List > adddirp1d | 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 8035 | . . 3 ⊢ (𝜑 → 1 ∈ ℂ) | |
3 | adddirp1d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 1, 2, 3 | adddird 8045 | . 2 ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + (1 · 𝐵))) |
5 | 3 | mulid2d 8038 | . . 3 ⊢ (𝜑 → (1 · 𝐵) = 𝐵) |
6 | 5 | oveq2d 5934 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (1 · 𝐵)) = ((𝐴 · 𝐵) + 𝐵)) |
7 | 4, 6 | eqtrd 2226 | 1 ⊢ (𝜑 → ((𝐴 + 1) · 𝐵) = ((𝐴 · 𝐵) + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2164 (class class class)co 5918 ℂcc 7870 1c1 7873 + caddc 7875 · cmul 7877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 ax-resscn 7964 ax-1cn 7965 ax-icn 7967 ax-addcl 7968 ax-mulcl 7970 ax-mulcom 7973 ax-mulass 7975 ax-distr 7976 ax-1rid 7979 ax-cnre 7983 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-iota 5215 df-fv 5262 df-ov 5921 |
This theorem is referenced by: modqvalp1 10414 hashxp 10897 fsumconst 11597 divalglemnqt 12061 pcexp 12447 mulgnnass 13227 cnfldmulg 14064 2lgsoddprmlem3d 15198 |
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