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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 10666 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = ((𝐴 · 𝐵) + (𝐶 · 𝐵))) |
5 | joinlmuladdmuld.4 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐵) + (𝐶 · 𝐵)) = 𝐷) | |
6 | 4, 5 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝐴 + 𝐶) · 𝐵) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 + caddc 10540 · cmul 10542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-addcl 10597 ax-mulcom 10601 ax-distr 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: 1p1times 10811 div4p1lem1div2 11893 ltdifltdiv 13205 discr1 13601 arisum 15215 bezoutlem3 15889 bezoutlem4 15890 mbfi1fseqlem4 24319 itgmulc2 24434 tangtx 25091 binom4 25428 axcontlem8 26757 int-rightdistd 40553 fmtnorec2lem 43724 joinlmuladdmuli 44894 |
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