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Mirrors > Home > MPE Home > Th. List > affineequivne | Structured version Visualization version GIF version |
Description: Equivalence between two ways of expressing 𝐴 as an affine combination of 𝐵 and 𝐶 if 𝐵 and 𝐶 are not equal. (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) |
affineequivne.d | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Ref | Expression |
---|---|
affineequivne | ⊢ (𝜑 → (𝐴 = (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) ↔ 𝐷 = ((𝐴 − 𝐵) / (𝐶 − 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 25116 | . 2 ⊢ (𝜑 → (𝐴 = (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) ↔ (𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)))) |
6 | eqcom 2779 | . . 3 ⊢ (((𝐴 − 𝐵) / (𝐶 − 𝐵)) = 𝐷 ↔ 𝐷 = ((𝐴 − 𝐵) / (𝐶 − 𝐵))) | |
7 | 1, 2 | subcld 10796 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) |
8 | 3, 2 | subcld 10796 | . . . 4 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ) |
9 | affineequivne.d | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
10 | 9 | necomd 3016 | . . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
11 | 3, 2, 10 | subne0d 10805 | . . . 4 ⊢ (𝜑 → (𝐶 − 𝐵) ≠ 0) |
12 | 7, 4, 8, 11 | divmul3d 11249 | . . 3 ⊢ (𝜑 → (((𝐴 − 𝐵) / (𝐶 − 𝐵)) = 𝐷 ↔ (𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)))) |
13 | 6, 12 | syl5rbbr 278 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐷 · (𝐶 − 𝐵)) ↔ 𝐷 = ((𝐴 − 𝐵) / (𝐶 − 𝐵)))) |
14 | 5, 13 | bitrd 271 | 1 ⊢ (𝜑 → (𝐴 = (((1 − 𝐷) · 𝐵) + (𝐷 · 𝐶)) ↔ 𝐷 = ((𝐴 − 𝐵) / (𝐶 − 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 (class class class)co 6974 ℂcc 10331 1c1 10334 + caddc 10336 · cmul 10338 − cmin 10668 / cdiv 11096 |
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 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
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 2016 df-mo 2547 df-eu 2584 df-clab 2753 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-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 |
This theorem is referenced by: affinecomb1 44082 |
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