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Theorem angpined 26747

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 26746	. 2 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))
8		1rp 12962	. . . . . 6 ⊢ 1 ∈ ℝ⁺
9		1re 11181	. . . . . . 7 ⊢ 1 ∈ ℝ
10		ax-1ne0 11144	. . . . . . 7 ⊢ 1 ≠ 0
11		rpneg 12992	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 1 ≠ 0) → (1 ∈ ℝ⁺ ↔ ¬ -1 ∈ ℝ⁺))
12	9, 10, 11	mp2an 692	. . . . . 6 ⊢ (1 ∈ ℝ⁺ ↔ ¬ -1 ∈ ℝ⁺)
13	8, 12	mpbi 230	. . . . 5 ⊢ ¬ -1 ∈ ℝ⁺
14	2, 3	subcld 11540	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
15	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴 − 𝐵) ∈ ℂ)
16	2, 3, 5	subne0d 11549	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0)
17	16	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴 − 𝐵) ≠ 0)
18		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
19	18	oveq1d 7405	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐶 − 𝐵) = (𝐴 − 𝐵))
20	15, 17, 19	diveq1bd 12013	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) = 1)
21	20	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) = 1)
22	21	negeqd 11422	. . . . . . . 8 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -((𝐶 − 𝐵) / (𝐴 − 𝐵)) = -1)
23		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺)
24	22, 23	eqeltrrd 2830	. . . . . . 7 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -1 ∈ ℝ⁺)
25	24	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) → (𝐶 = 𝐴 → -1 ∈ ℝ⁺))
26	25	necon3bd 2940	. . . . 5 ⊢ ((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) → (¬ -1 ∈ ℝ⁺ → 𝐶 ≠ 𝐴))
27	13, 26	mpi 20	. . . 4 ⊢ ((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) → 𝐶 ≠ 𝐴)
28	27	ex 412	. . 3 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ → 𝐶 ≠ 𝐴))
29		necom 2979	. . 3 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐴 ≠ 𝐶)
30	28, 29	imbitrdi 251	. 2 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ → 𝐴 ≠ 𝐶))
31	7, 30	sylbird 260	1 ⊢ (𝜑 → (((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π → 𝐴 ≠ 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∖ cdif 3914 {csn 4592 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 − cmin 11412 -cneg 11413 / cdiv 11842 ℝ⁺crp 12958 ℑcim 15071 πcpi 16039 logclog 26470

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040 df-sin 16042 df-cos 16043 df-pi 16045 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775 df-log 26472

This theorem is referenced by: angpieqvd 26748

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