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Mirrors > Home > MPE Home > Th. List > affineequiv3 | Structured version Visualization version GIF version |
Description: Equivalence between two ways of expressing 𝐴 as an affine combination of 𝐵 and 𝐶. (Contributed by AV, 22-Jan-2023.) |
Ref | Expression |
---|---|
affineequiv.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
affineequiv.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
affineequiv.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
affineequiv.d | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
Ref | Expression |
---|---|
affineequiv3 | ⊢ (𝜑 → (𝐴 = (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) ↔ (𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1cnd 10629 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
2 | affineequiv.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
3 | 1, 2 | subcld 10990 | . . . . 5 ⊢ (𝜑 → (1 − 𝐷) ∈ ℂ) |
4 | affineequiv.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
5 | 3, 4 | mulcld 10654 | . . . 4 ⊢ (𝜑 → ((1 − 𝐷) · 𝐵) ∈ ℂ) |
6 | affineequiv.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
7 | 2, 6 | mulcld 10654 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝐶) ∈ ℂ) |
8 | 5, 7 | addcomd 10835 | . . 3 ⊢ (𝜑 → (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) = ((𝐷 · 𝐶) + ((1 − 𝐷) · 𝐵))) |
9 | 8 | eqeq2d 2831 | . 2 ⊢ (𝜑 → (𝐴 = (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) ↔ 𝐴 = ((𝐷 · 𝐶) + ((1 − 𝐷) · 𝐵)))) |
10 | affineequiv.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
11 | 6, 10, 4, 2 | affineequiv 25397 | . 2 ⊢ (𝜑 → (𝐴 = ((𝐷 · 𝐶) + ((1 − 𝐷) · 𝐵)) ↔ (𝐵 − 𝐴) = (𝐷 · (𝐵 − 𝐶)))) |
12 | 10, 4 | negsubdi2d 11006 | . . . . 5 ⊢ (𝜑 → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) |
13 | 12 | eqcomd 2826 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) = -(𝐴 − 𝐵)) |
14 | 13 | eqeq1d 2822 | . . 3 ⊢ (𝜑 → ((𝐵 − 𝐴) = (𝐷 · (𝐵 − 𝐶)) ↔ -(𝐴 − 𝐵) = (𝐷 · (𝐵 − 𝐶)))) |
15 | 6, 4 | negsubdi2d 11006 | . . . . . . 7 ⊢ (𝜑 → -(𝐶 − 𝐵) = (𝐵 − 𝐶)) |
16 | 15 | eqcomd 2826 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐶) = -(𝐶 − 𝐵)) |
17 | 16 | oveq2d 7165 | . . . . 5 ⊢ (𝜑 → (𝐷 · (𝐵 − 𝐶)) = (𝐷 · -(𝐶 − 𝐵))) |
18 | 6, 4 | subcld 10990 | . . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ) |
19 | 2, 18 | mulneg2d 11087 | . . . . 5 ⊢ (𝜑 → (𝐷 · -(𝐶 − 𝐵)) = -(𝐷 · (𝐶 − 𝐵))) |
20 | 17, 19 | eqtrd 2855 | . . . 4 ⊢ (𝜑 → (𝐷 · (𝐵 − 𝐶)) = -(𝐷 · (𝐶 − 𝐵))) |
21 | 20 | eqeq2d 2831 | . . 3 ⊢ (𝜑 → (-(𝐴 − 𝐵) = (𝐷 · (𝐵 − 𝐶)) ↔ -(𝐴 − 𝐵) = -(𝐷 · (𝐶 − 𝐵)))) |
22 | 10, 4 | subcld 10990 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
23 | 2, 18 | mulcld 10654 | . . . 4 ⊢ (𝜑 → (𝐷 · (𝐶 − 𝐵)) ∈ ℂ) |
24 | 22, 23 | neg11ad 10986 | . . 3 ⊢ (𝜑 → (-(𝐴 − 𝐵) = -(𝐷 · (𝐶 − 𝐵)) ↔ (𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)))) |
25 | 14, 21, 24 | 3bitrd 307 | . 2 ⊢ (𝜑 → ((𝐵 − 𝐴) = (𝐷 · (𝐵 − 𝐶)) ↔ (𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)))) |
26 | 9, 11, 25 | 3bitrd 307 | 1 ⊢ (𝜑 → (𝐴 = (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) ↔ (𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 (class class class)co 7149 ℂcc 10528 1c1 10531 + caddc 10533 · cmul 10535 − cmin 10863 -cneg 10864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-ltxr 10673 df-sub 10865 df-neg 10866 |
This theorem is referenced by: affineequiv4 25400 affineequivne 25401 |
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