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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 25553 | . 2 ⊢ (𝜑 → (𝐵 = ((𝐷 · 𝐴) + ((1 − 𝐷) · 𝐶)) ↔ (𝐶 − 𝐵) = (𝐷 · (𝐶 − 𝐴)))) |
6 | 3, 1 | subcld 11068 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℂ) |
7 | 3, 2 | subcld 11068 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ) |
8 | 4, 6 | mulcld 10732 | . . 3 ⊢ (𝜑 → (𝐷 · (𝐶 − 𝐴)) ∈ ℂ) |
9 | 6, 7, 8 | subcanad 11111 | . 2 ⊢ (𝜑 → (((𝐶 − 𝐴) − (𝐶 − 𝐵)) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) ↔ (𝐶 − 𝐵) = (𝐷 · (𝐶 − 𝐴)))) |
10 | 3, 1, 2 | nnncan1d 11102 | . . 3 ⊢ (𝜑 → ((𝐶 − 𝐴) − (𝐶 − 𝐵)) = (𝐵 − 𝐴)) |
11 | 1cnd 10707 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
12 | 11, 4, 6 | subdird 11168 | . . . 4 ⊢ (𝜑 → ((1 − 𝐷) · (𝐶 − 𝐴)) = ((1 · (𝐶 − 𝐴)) − (𝐷 · (𝐶 − 𝐴)))) |
13 | 6 | mulid2d 10730 | . . . . 5 ⊢ (𝜑 → (1 · (𝐶 − 𝐴)) = (𝐶 − 𝐴)) |
14 | 13 | oveq1d 7179 | . . . 4 ⊢ (𝜑 → ((1 · (𝐶 − 𝐴)) − (𝐷 · (𝐶 − 𝐴))) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴)))) |
15 | 12, 14 | eqtr2d 2774 | . . 3 ⊢ (𝜑 → ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) = ((1 − 𝐷) · (𝐶 − 𝐴))) |
16 | 10, 15 | eqeq12d 2754 | . 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 1542 ∈ wcel 2113 (class class class)co 7164 ℂcc 10606 1c1 10609 + caddc 10611 · cmul 10613 − cmin 10941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 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 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-ltxr 10751 df-sub 10943 |
This theorem is referenced by: chordthmlem4 25565 |
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