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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 9946 | . 2 ⊢ (⊤ → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
10 | 9 | trud 1484 | 1 ⊢ ((𝐴 + 𝐶) · 𝐵) = 𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 + caddc 9818 · cmul 9820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-addcl 9875 ax-mulcom 9879 ax-distr 9882 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: (None) |
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