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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 11286 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵))) |
| 5 | joinlmuladdmuld.4 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) | |
| 6 | 4, 5 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 + caddc 11158 · cmul 11160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-addcl 11215 ax-mulcom 11219 ax-distr 11222 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: 1p1times 11432 div4p1lem1div2 12521 ltdifltdiv 13874 discr1 14278 arisum 15896 bezoutlem3 16578 bezoutlem4 16579 mbfi1fseqlem4 25753 itgmulc2 25869 tangtx 26547 binom4 26893 axcontlem8 28986 zringfrac 33582 constrrtcclem 33775 int-rightdistd 44193 fmtnorec2lem 47529 joinlmuladdmuli 49292 |
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