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Mirrors > Home > MPE Home > Th. List > Mathboxes > joinlmuladdmuli | Structured version Visualization version GIF version |
Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.) |
Ref | Expression |
---|---|
joinlmuladdmuli.1 | ⊢ 𝐴 ∈ ℂ |
joinlmuladdmuli.2 | ⊢ 𝐵 ∈ ℂ |
joinlmuladdmuli.3 | ⊢ 𝐶 ∈ ℂ |
joinlmuladdmuli.4 | ⊢ ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷 |
Ref | Expression |
---|---|
joinlmuladdmuli | ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joinlmuladdmuli.1 | . . . 4 ⊢ 𝐴 ∈ ℂ | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → 𝐴 ∈ ℂ) |
3 | joinlmuladdmuli.2 | . . . 4 ⊢ 𝐵 ∈ ℂ | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → 𝐵 ∈ ℂ) |
5 | joinlmuladdmuli.3 | . . . 4 ⊢ 𝐶 ∈ ℂ | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → 𝐶 ∈ ℂ) |
7 | joinlmuladdmuli.4 | . . . 4 ⊢ ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷 | |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) |
9 | 2, 4, 6, 8 | joinlmuladdmuld 11140 | . 2 ⊢ (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
10 | 9 | mptru 1548 | 1 ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 (class class class)co 7351 ℂcc 11007 + caddc 11012 · cmul 11014 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-addcl 11069 ax-mulcom 11073 ax-distr 11076 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-ov 7354 |
This theorem is referenced by: (None) |
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