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Mirrors > Home > MPE Home > Th. List > joinlmuladdmuld | 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 |
---|---|
joinlmuladdmuld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
joinlmuladdmuld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
joinlmuladdmuld.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
joinlmuladdmuld.4 | ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) |
Ref | Expression |
---|---|
joinlmuladdmuld | ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joinlmuladdmuld.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | joinlmuladdmuld.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
3 | joinlmuladdmuld.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 1, 2, 3 | adddird 9944 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵))) |
5 | joinlmuladdmuld.4 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) | |
6 | 4, 5 | eqtrd 2644 | 1 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ 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: div4p1lem1div2 11164 fmtnorec2lem 39992 joinlmuladdmuli 42328 |
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