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Theorem angpined 26961

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 26960	. 2 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))
8		1rp 13020	. . . . . 6 ⊢ 1 ∈ ℝ⁺
9		1re 11208	. . . . . . 7 ⊢ 1 ∈ ℝ
10		ax-1ne0 11169	. . . . . . 7 ⊢ 1 ≠ 0
11		rpneg 13050	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 1 ≠ 0) → (1 ∈ ℝ⁺ ↔ ¬ -1 ∈ ℝ⁺))
12	9, 10, 11	mp2an 704	. . . . . 6 ⊢ (1 ∈ ℝ⁺ ↔ ¬ -1 ∈ ℝ⁺)
13	8, 12	mpbi 233	. . . . 5 ⊢ ¬ -1 ∈ ℝ⁺
14	2, 3	subcld 11569	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
15	14	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴 − 𝐵) ∈ ℂ)
16	2, 3, 5	subne0d 11578	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0)
17	16	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴 − 𝐵) ≠ 0)
18		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
19	18	oveq1d 7426	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐶 − 𝐵) = (𝐴 − 𝐵))
20	15, 17, 19	diveq1bd 12039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) = 1)
21	20	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) = 1)
22	21	negeqd 11451	. . . . . . . 8 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -((𝐶 − 𝐵) / (𝐴 − 𝐵)) = -1)
23		simplr 780	. . . . . . . 8 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺)
24	22, 23	eqeltrrd 2870	. . . . . . 7 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -1 ∈ ℝ⁺)
25	24	ex 417	. . . . . 6 ⊢ ((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) → (𝐶 = 𝐴 → -1 ∈ ℝ⁺))
26	25	necon3bd 2978	. . . . 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 417	. . 3 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ → 𝐶 ≠ 𝐴))
29		necom 3017	. . 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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∖ cdif 3908 {csn 4592 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 − cmin 11441 -cneg 11442 / cdiv 11871 ℝ⁺crp 13016 ℑcim 15149 πcpi 16120 logclog 26685

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3375 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ioc 13377 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15104 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-limsup 15522 df-clim 15539 df-rlim 15540 df-sum 15738 df-ef 16121 df-sin 16123 df-cos 16124 df-pi 16126 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-lp 23262 df-perf 23263 df-cn 23353 df-cnp 23354 df-haus 23441 df-tx 23688 df-hmeo 23881 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-xms 24446 df-ms 24447 df-tms 24448 df-cncf 25006 df-limc 25994 df-dv 25995 df-log 26687

This theorem is referenced by: angpieqvd 26962

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