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Mirrors > Home > MPE Home > Th. List > affineequiv2 | Structured version Visualization version GIF version |
Description: Equivalence between two ways of expressing 𝐵 as an affine combination of 𝐴 and 𝐶. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
affineequiv.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
affineequiv.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
affineequiv.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
affineequiv.d | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
affineequiv2 | ⊢ (𝜑 → (𝐵 = ((𝐷 · 𝐴) + ((1 − 𝐷) · 𝐶)) ↔ (𝐵 − 𝐴) = ((1 − 𝐷) · (𝐶 − 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | affineequiv.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | affineequiv.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | affineequiv.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | affineequiv.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
5 | 1, 2, 3, 4 | affineequiv 25012 | . 2 ⊢ (𝜑 → (𝐵 = ((𝐷 · 𝐴) + ((1 − 𝐷) · 𝐶)) ↔ (𝐶 − 𝐵) = (𝐷 · (𝐶 − 𝐴)))) |
6 | 3, 1 | subcld 10736 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℂ) |
7 | 3, 2 | subcld 10736 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ) |
8 | 4, 6 | mulcld 10399 | . . 3 ⊢ (𝜑 → (𝐷 · (𝐶 − 𝐴)) ∈ ℂ) |
9 | 6, 7, 8 | subcanad 10779 | . 2 ⊢ (𝜑 → (((𝐶 − 𝐴) − (𝐶 − 𝐵)) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) ↔ (𝐶 − 𝐵) = (𝐷 · (𝐶 − 𝐴)))) |
10 | 3, 1, 2 | nnncan1d 10770 | . . 3 ⊢ (𝜑 → ((𝐶 − 𝐴) − (𝐶 − 𝐵)) = (𝐵 − 𝐴)) |
11 | 1cnd 10373 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
12 | 11, 4, 6 | subdird 10835 | . . . 4 ⊢ (𝜑 → ((1 − 𝐷) · (𝐶 − 𝐴)) = ((1 · (𝐶 − 𝐴)) − (𝐷 · (𝐶 − 𝐴)))) |
13 | 6 | mulid2d 10397 | . . . . 5 ⊢ (𝜑 → (1 · (𝐶 − 𝐴)) = (𝐶 − 𝐴)) |
14 | 13 | oveq1d 6939 | . . . 4 ⊢ (𝜑 → ((1 · (𝐶 − 𝐴)) − (𝐷 · (𝐶 − 𝐴))) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴)))) |
15 | 12, 14 | eqtr2d 2815 | . . 3 ⊢ (𝜑 → ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) = ((1 − 𝐷) · (𝐶 − 𝐴))) |
16 | 10, 15 | eqeq12d 2793 | . 2 ⊢ (𝜑 → (((𝐶 − 𝐴) − (𝐶 − 𝐵)) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) ↔ (𝐵 − 𝐴) = ((1 − 𝐷) · (𝐶 − 𝐴)))) |
17 | 5, 9, 16 | 3bitr2d 299 | 1 ⊢ (𝜑 → (𝐵 = ((𝐷 · 𝐴) + ((1 − 𝐷) · 𝐶)) ↔ (𝐵 − 𝐴) = ((1 − 𝐷) · (𝐶 − 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 (class class class)co 6924 ℂcc 10272 1c1 10275 + caddc 10277 · cmul 10279 − cmin 10608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-ltxr 10418 df-sub 10610 |
This theorem is referenced by: chordthmlem4 25024 |
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