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Mirrors > Home > MPE Home > Th. List > Mathboxes > submuladdmuld | Structured version Visualization version GIF version |
Description: Transformation of a sum of a product of a difference and a product with the subtrahend of the difference. (Contributed by AV, 2-Feb-2023.) |
Ref | Expression |
---|---|
submuladdmuld.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
submuladdmuld.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
submuladdmuld.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
submuladdmuld.d | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
submuladdmuld | ⊢ (𝜑 → (((𝐴 − 𝐵) · 𝐶) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + (𝐵 · (𝐷 − 𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submuladdmuld.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | submuladdmuld.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | submuladdmuld.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | 1, 2, 3 | subdird 11097 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶))) |
5 | 4 | oveq1d 7171 | . 2 ⊢ (𝜑 → (((𝐴 − 𝐵) · 𝐶) + (𝐵 · 𝐷)) = (((𝐴 · 𝐶) − (𝐵 · 𝐶)) + (𝐵 · 𝐷))) |
6 | 1, 3 | mulcld 10661 | . . 3 ⊢ (𝜑 → (𝐴 · 𝐶) ∈ ℂ) |
7 | 2, 3 | mulcld 10661 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐶) ∈ ℂ) |
8 | submuladdmuld.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
9 | 2, 8 | mulcld 10661 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐷) ∈ ℂ) |
10 | 6, 7, 9 | subadd23d 11019 | . 2 ⊢ (𝜑 → (((𝐴 · 𝐶) − (𝐵 · 𝐶)) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + ((𝐵 · 𝐷) − (𝐵 · 𝐶)))) |
11 | 2, 8, 3 | subdid 11096 | . . . 4 ⊢ (𝜑 → (𝐵 · (𝐷 − 𝐶)) = ((𝐵 · 𝐷) − (𝐵 · 𝐶))) |
12 | 11 | eqcomd 2827 | . . 3 ⊢ (𝜑 → ((𝐵 · 𝐷) − (𝐵 · 𝐶)) = (𝐵 · (𝐷 − 𝐶))) |
13 | 12 | oveq2d 7172 | . 2 ⊢ (𝜑 → ((𝐴 · 𝐶) + ((𝐵 · 𝐷) − (𝐵 · 𝐶))) = ((𝐴 · 𝐶) + (𝐵 · (𝐷 − 𝐶)))) |
14 | 5, 10, 13 | 3eqtrd 2860 | 1 ⊢ (𝜑 → (((𝐴 − 𝐵) · 𝐶) + (𝐵 · 𝐷)) = ((𝐴 · 𝐶) + (𝐵 · (𝐷 − 𝐶)))) |
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 − cmin 10870 |
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-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 |
This theorem is referenced by: rrx2vlinest 44777 |
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