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Mirrors > Home > MPE Home > Th. List > affineequiv4 | 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 |
---|---|
affineequiv4 | ⊢ (𝜑 → (𝐴 = (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) ↔ 𝐴 = ((𝐷 · (𝐶 − 𝐵)) + 𝐵))) |
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 | affineequiv3 25094 | . 2 ⊢ (𝜑 → (𝐴 = (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) ↔ (𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)))) |
6 | 3, 2 | subcld 10790 | . . . . 5 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ) |
7 | 4, 6 | mulcld 10452 | . . . 4 ⊢ (𝜑 → (𝐷 · (𝐶 − 𝐵)) ∈ ℂ) |
8 | 1, 2, 7 | subadd2d 10809 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)) ↔ ((𝐷 · (𝐶 − 𝐵)) + 𝐵) = 𝐴)) |
9 | eqcom 2779 | . . 3 ⊢ (((𝐷 · (𝐶 − 𝐵)) + 𝐵) = 𝐴 ↔ 𝐴 = ((𝐷 · (𝐶 − 𝐵)) + 𝐵)) | |
10 | 8, 9 | syl6bb 279 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)) ↔ 𝐴 = ((𝐷 · (𝐶 − 𝐵)) + 𝐵))) |
11 | 5, 10 | bitrd 271 | 1 ⊢ (𝜑 → (𝐴 = (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) ↔ 𝐴 = ((𝐷 · (𝐶 − 𝐵)) + 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2048 (class class class)co 6970 ℂcc 10325 1c1 10328 + caddc 10330 · cmul 10332 − cmin 10662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-po 5319 df-so 5320 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-ltxr 10471 df-sub 10664 df-neg 10665 |
This theorem is referenced by: affinecomb1 43997 |
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