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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 25409 | . 2 ⊢ (𝜑 → (𝐵 = ((𝐷 · 𝐴) + ((1 − 𝐷) · 𝐶)) ↔ (𝐶 − 𝐵) = (𝐷 · (𝐶 − 𝐴)))) |
6 | 3, 1 | subcld 10986 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℂ) |
7 | 3, 2 | subcld 10986 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ) |
8 | 4, 6 | mulcld 10650 | . . 3 ⊢ (𝜑 → (𝐷 · (𝐶 − 𝐴)) ∈ ℂ) |
9 | 6, 7, 8 | subcanad 11029 | . 2 ⊢ (𝜑 → (((𝐶 − 𝐴) − (𝐶 − 𝐵)) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) ↔ (𝐶 − 𝐵) = (𝐷 · (𝐶 − 𝐴)))) |
10 | 3, 1, 2 | nnncan1d 11020 | . . 3 ⊢ (𝜑 → ((𝐶 − 𝐴) − (𝐶 − 𝐵)) = (𝐵 − 𝐴)) |
11 | 1cnd 10625 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
12 | 11, 4, 6 | subdird 11086 | . . . 4 ⊢ (𝜑 → ((1 − 𝐷) · (𝐶 − 𝐴)) = ((1 · (𝐶 − 𝐴)) − (𝐷 · (𝐶 − 𝐴)))) |
13 | 6 | mulid2d 10648 | . . . . 5 ⊢ (𝜑 → (1 · (𝐶 − 𝐴)) = (𝐶 − 𝐴)) |
14 | 13 | oveq1d 7150 | . . . 4 ⊢ (𝜑 → ((1 · (𝐶 − 𝐴)) − (𝐷 · (𝐶 − 𝐴))) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴)))) |
15 | 12, 14 | eqtr2d 2834 | . . 3 ⊢ (𝜑 → ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) = ((1 − 𝐷) · (𝐶 − 𝐴))) |
16 | 10, 15 | eqeq12d 2814 | . 2 ⊢ (𝜑 → (((𝐶 − 𝐴) − (𝐶 − 𝐵)) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) ↔ (𝐵 − 𝐴) = ((1 − 𝐷) · (𝐶 − 𝐴)))) |
17 | 5, 9, 16 | 3bitr2d 310 | 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 209 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 |
This theorem is referenced by: chordthmlem4 25421 |
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