angpined - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > angpined		Structured version Visualization version GIF version

Theorem angpined 27099

Description: If the angle at ABC is π, then 𝐴 is not equal to 𝐶. (Contributed by David Moews, 28-Feb-2017.)

**Hypotheses**
Ref	Expression
angpieqvd.angdef	⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
angpieqvd.A	⊢ (𝜑 → 𝐴 ∈ ℂ)
angpieqvd.B	⊢ (𝜑 → 𝐵 ∈ ℂ)
angpieqvd.C	⊢ (𝜑 → 𝐶 ∈ ℂ)
angpieqvd.AneB	⊢ (𝜑 → 𝐴 ≠ 𝐵)
angpieqvd.BneC	⊢ (𝜑 → 𝐵 ≠ 𝐶)

**Assertion**
Ref	Expression
angpined	⊢ (𝜑 → (((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π → 𝐴 ≠ 𝐶))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦 𝑥,𝐶,𝑦 Allowed substitution hints: 𝜑(𝑥, 𝑦) 𝐹(𝑥, 𝑦)

**Proof of Theorem angpined**
Step	Hyp	Ref	Expression
1		angpieqvd.angdef	. . 3 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
2		angpieqvd.A	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		angpieqvd.B	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ)
4		angpieqvd.C	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5		angpieqvd.AneB	. . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
6		angpieqvd.BneC	. . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
7	1, 2, 3, 4, 5, 6	angpieqvdlem2 27098	. 2 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))
8		1rp 13071	. . . . . 6 ⊢ 1 ∈ ℝ⁺
9		1re 11257	. . . . . . 7 ⊢ 1 ∈ ℝ
10		ax-1ne0 11218	. . . . . . 7 ⊢ 1 ≠ 0
11		rpneg 13101	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 1 ≠ 0) → (1 ∈ ℝ⁺ ↔ ¬ -1 ∈ ℝ⁺))
12	9, 10, 11	mp2an 705	. . . . . 6 ⊢ (1 ∈ ℝ⁺ ↔ ¬ -1 ∈ ℝ⁺)
13	8, 12	mpbi 233	. . . . 5 ⊢ ¬ -1 ∈ ℝ⁺
14	2, 3	subcld 11618	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
15	14	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴 − 𝐵) ∈ ℂ)
16	2, 3, 5	subne0d 11628	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0)
17	16	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴 − 𝐵) ≠ 0)
18		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
19	18	oveq1d 7431	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐶 − 𝐵) = (𝐴 − 𝐵))
20	15, 17, 19	diveq1bd 12088	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) = 1)
21	20	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) = 1)
22	21	negeqd 11500	. . . . . . . 8 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -((𝐶 − 𝐵) / (𝐴 − 𝐵)) = -1)
23		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺)
24	22, 23	eqeltrrd 2861	. . . . . . 7 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -1 ∈ ℝ⁺)
25	24	ex 418	. . . . . 6 ⊢ ((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) → (𝐶 = 𝐴 → -1 ∈ ℝ⁺))
26	25	necon3bd 2969	. . . . 5 ⊢ ((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) → (¬ -1 ∈ ℝ⁺ → 𝐶 ≠ 𝐴))
27	13, 26	mpi 21	. . . 4 ⊢ ((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) → 𝐶 ≠ 𝐴)
28	27	ex 418	. . 3 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ → 𝐶 ≠ 𝐴))
29		necom 3008	. . 3 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐴 ≠ 𝐶)
30	28, 29	imbitrdi 254	. 2 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ → 𝐴 ≠ 𝐶))
31	7, 30	sylbird 263	1 ⊢ (𝜑 → (((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π → 𝐴 ≠ 𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∖ cdif 3896 {csn 4584 ‘cfv 6535 (class class class)co 7416 ∈ cmpo 7418 ℂcc 11147 ℝcr 11148 0cc0 11149 1c1 11150 − cmin 11490 -cneg 11491 / cdiv 11920 ℝ⁺crp 13067 ℑcim 15210 πcpi 16177 logclog 26823

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-inf2 9627 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9489 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-q 13023 df-rp 13068 df-xneg 13188 df-xadd 13189 df-xmul 13190 df-ioo 13427 df-ioc 13428 df-ico 13429 df-icc 13430 df-fz 13587 df-fzo 13735 df-fl 13878 df-mod 13956 df-seq 14091 df-exp 14151 df-fac 14363 df-bc 14392 df-hash 14420 df-shft 15165 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-limsup 15583 df-clim 15600 df-rlim 15601 df-sum 15799 df-ef 16178 df-sin 16180 df-cos 16181 df-pi 16183 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-rest 17532 df-topn 17533 df-0g 17551 df-gsum 17552 df-topgen 17553 df-pt 17554 df-prds 17557 df-xrs 17613 df-qtop 17618 df-imas 17619 df-xps 17621 df-mre 17695 df-mrc 17696 df-acs 17698 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-submnd 18918 df-mulg 19217 df-cntz 19470 df-cmn 19935 df-psmet 21609 df-xmet 21610 df-met 21611 df-bl 21612 df-mopn 21613 df-fbas 21614 df-fg 21615 df-cnfld 21618 df-top 23151 df-topon 23168 df-topsp 23190 df-bases 23203 df-cld 23276 df-ntr 23277 df-cls 23278 df-nei 23355 df-lp 23393 df-perf 23394 df-cn 23484 df-cnp 23485 df-haus 23572 df-tx 23820 df-hmeo 24013 df-fil 24104 df-fm 24196 df-flim 24197 df-flf 24198 df-xms 24578 df-ms 24579 df-tms 24580 df-cncf 25138 df-limc 26125 df-dv 26126 df-log 26825

This theorem is used by: angpieqvd 27100

W3C validator