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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 11256 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ) | |
| 2 | affineequiv.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
| 3 | 1, 2 | subcld 11620 | . . . . 5 ⊢ (𝜑 → (1 − 𝐷) ∈ ℂ) |
| 4 | affineequiv.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 5 | 3, 4 | mulcld 11281 | . . . 4 ⊢ (𝜑 → ((1 − 𝐷) · 𝐵) ∈ ℂ) |
| 6 | affineequiv.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
| 7 | 2, 6 | mulcld 11281 | . . . 4 ⊢ (𝜑 → (𝐷 · 𝐶) ∈ ℂ) |
| 8 | 5, 7 | addcomd 11463 | . . 3 ⊢ (𝜑 → (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) = ((𝐷 · 𝐶) + ((1 − 𝐷) · 𝐵))) |
| 9 | 8 | eqeq2d 2748 | . 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 26866 | . 2 ⊢ (𝜑 → (𝐴 = ((𝐷 · 𝐶) + ((1 − 𝐷) · 𝐵)) ↔ (𝐵 − 𝐴) = (𝐷 · (𝐵 − 𝐶)))) |
| 12 | 10, 4 | negsubdi2d 11636 | . . . . 5 ⊢ (𝜑 → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) |
| 13 | 12 | eqcomd 2743 | . . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) = -(𝐴 − 𝐵)) |
| 14 | 13 | eqeq1d 2739 | . . 3 ⊢ (𝜑 → ((𝐵 − 𝐴) = (𝐷 · (𝐵 − 𝐶)) ↔ -(𝐴 − 𝐵) = (𝐷 · (𝐵 − 𝐶)))) |
| 15 | 6, 4 | negsubdi2d 11636 | . . . . . . 7 ⊢ (𝜑 → -(𝐶 − 𝐵) = (𝐵 − 𝐶)) |
| 16 | 15 | eqcomd 2743 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐶) = -(𝐶 − 𝐵)) |
| 17 | 16 | oveq2d 7447 | . . . . 5 ⊢ (𝜑 → (𝐷 · (𝐵 − 𝐶)) = (𝐷 · -(𝐶 − 𝐵))) |
| 18 | 6, 4 | subcld 11620 | . . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ) |
| 19 | 2, 18 | mulneg2d 11717 | . . . . 5 ⊢ (𝜑 → (𝐷 · -(𝐶 − 𝐵)) = -(𝐷 · (𝐶 − 𝐵))) |
| 20 | 17, 19 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → (𝐷 · (𝐵 − 𝐶)) = -(𝐷 · (𝐶 − 𝐵))) |
| 21 | 20 | eqeq2d 2748 | . . 3 ⊢ (𝜑 → (-(𝐴 − 𝐵) = (𝐷 · (𝐵 − 𝐶)) ↔ -(𝐴 − 𝐵) = -(𝐷 · (𝐶 − 𝐵)))) |
| 22 | 10, 4 | subcld 11620 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
| 23 | 2, 18 | mulcld 11281 | . . . 4 ⊢ (𝜑 → (𝐷 · (𝐶 − 𝐵)) ∈ ℂ) |
| 24 | 22, 23 | neg11ad 11616 | . . 3 ⊢ (𝜑 → (-(𝐴 − 𝐵) = -(𝐷 · (𝐶 − 𝐵)) ↔ (𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)))) |
| 25 | 14, 21, 24 | 3bitrd 305 | . 2 ⊢ (𝜑 → ((𝐵 − 𝐴) = (𝐷 · (𝐵 − 𝐶)) ↔ (𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)))) |
| 26 | 9, 11, 25 | 3bitrd 305 | 1 ⊢ (𝜑 → (𝐴 = (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) ↔ (𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 -cneg 11493 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 |
| This theorem is referenced by: affineequiv4 26869 affineequivne 26870 |
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