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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 11313 | . 2 ⊢ (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
10 | 9 | mptru 1544 | 1 ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊤wtru 1538 ∈ wcel 2103 (class class class)co 7445 ℂcc 11178 + caddc 11183 · cmul 11185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 ax-addcl 11240 ax-mulcom 11244 ax-distr 11247 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-iota 6524 df-fv 6580 df-ov 7448 |
This theorem is referenced by: (None) |
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