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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 11284 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵))) |
5 | joinlmuladdmuld.4 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) | |
6 | 4, 5 | eqtrd 2775 | 1 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 + caddc 11156 · cmul 11158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-addcl 11213 ax-mulcom 11217 ax-distr 11220 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: 1p1times 11430 div4p1lem1div2 12519 ltdifltdiv 13871 discr1 14275 arisum 15893 bezoutlem3 16575 bezoutlem4 16576 mbfi1fseqlem4 25768 itgmulc2 25884 tangtx 26562 binom4 26908 axcontlem8 29001 zringfrac 33562 constrrtcclem 33740 int-rightdistd 44170 fmtnorec2lem 47467 joinlmuladdmuli 49004 |
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