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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 24553 | . 2 ⊢ (𝜑 → (𝐵 = ((𝐷 · 𝐴) + ((1 − 𝐷) · 𝐶)) ↔ (𝐶 − 𝐵) = (𝐷 · (𝐶 − 𝐴)))) |
6 | 3, 1 | subcld 10392 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℂ) |
7 | 3, 2 | subcld 10392 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ) |
8 | 4, 6 | mulcld 10060 | . . 3 ⊢ (𝜑 → (𝐷 · (𝐶 − 𝐴)) ∈ ℂ) |
9 | 6, 7, 8 | subcanad 10435 | . 2 ⊢ (𝜑 → (((𝐶 − 𝐴) − (𝐶 − 𝐵)) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) ↔ (𝐶 − 𝐵) = (𝐷 · (𝐶 − 𝐴)))) |
10 | 3, 1, 2 | nnncan1d 10426 | . . 3 ⊢ (𝜑 → ((𝐶 − 𝐴) − (𝐶 − 𝐵)) = (𝐵 − 𝐴)) |
11 | 1cnd 10056 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℂ) | |
12 | 11, 4, 6 | subdird 10487 | . . . 4 ⊢ (𝜑 → ((1 − 𝐷) · (𝐶 − 𝐴)) = ((1 · (𝐶 − 𝐴)) − (𝐷 · (𝐶 − 𝐴)))) |
13 | 6 | mulid2d 10058 | . . . . 5 ⊢ (𝜑 → (1 · (𝐶 − 𝐴)) = (𝐶 − 𝐴)) |
14 | 13 | oveq1d 6665 | . . . 4 ⊢ (𝜑 → ((1 · (𝐶 − 𝐴)) − (𝐷 · (𝐶 − 𝐴))) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴)))) |
15 | 12, 14 | eqtr2d 2657 | . . 3 ⊢ (𝜑 → ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) = ((1 − 𝐷) · (𝐶 − 𝐴))) |
16 | 10, 15 | eqeq12d 2637 | . 2 ⊢ (𝜑 → (((𝐶 − 𝐴) − (𝐶 − 𝐵)) = ((𝐶 − 𝐴) − (𝐷 · (𝐶 − 𝐴))) ↔ (𝐵 − 𝐴) = ((1 − 𝐷) · (𝐶 − 𝐴)))) |
17 | 5, 9, 16 | 3bitr2d 296 | 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 196 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 |
This theorem is referenced by: chordthmlem4 24562 |
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