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Theorem angpined 27063

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 27062	. 2 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))
8		1rp 13046	. . . . . 6 ⊢ 1 ∈ ℝ⁺
9		1re 11233	. . . . . . 7 ⊢ 1 ∈ ℝ
10		ax-1ne0 11194	. . . . . . 7 ⊢ 1 ≠ 0
11		rpneg 13076	. . . . . . 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 11594	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
15	14	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴 − 𝐵) ∈ ℂ)
16	2, 3, 5	subne0d 11603	. . . . . . . . . . . 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 12064	. . . . . . . . . 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 11476	. . . . . . . 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 2863	. . . . . . 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 2971	. . . . 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 3010	. . 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 2957 ∖ cdif 3899 {csn 4587 ‘cfv 6537 (class class class)co 7416 ∈ cmpo 7418 ℂcc 11123 ℝcr 11124 0cc0 11125 1c1 11126 − cmin 11466 -cneg 11467 / cdiv 11896 ℝ⁺crp 13042 ℑcim 15185 πcpi 16154 logclog 26787

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-fac 14338 df-bc 14367 df-hash 14395 df-shft 15140 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-ef 16155 df-sin 16157 df-cos 16158 df-pi 16160 df-struct 17241 df-sets 17258 df-slot 17276 df-ndx 17288 df-base 17304 df-ress 17325 df-plusg 17357 df-mulr 17358 df-starv 17359 df-sca 17360 df-vsca 17361 df-ip 17362 df-tset 17363 df-ple 17364 df-ds 17366 df-unif 17367 df-hom 17368 df-cco 17369 df-rest 17509 df-topn 17510 df-0g 17528 df-gsum 17529 df-topgen 17530 df-pt 17531 df-prds 17534 df-xrs 17590 df-qtop 17595 df-imas 17596 df-xps 17598 df-mre 17672 df-mrc 17673 df-acs 17675 df-mgm 18732 df-sgrp 18821 df-mnd 18837 df-submnd 18891 df-mulg 19190 df-cntz 19443 df-cmn 19908 df-psmet 21576 df-xmet 21577 df-met 21578 df-bl 21579 df-mopn 21580 df-fbas 21581 df-fg 21582 df-cnfld 21585 df-top 23118 df-topon 23135 df-topsp 23157 df-bases 23170 df-cld 23243 df-ntr 23244 df-cls 23245 df-nei 23322 df-lp 23360 df-perf 23361 df-cn 23451 df-cnp 23452 df-haus 23539 df-tx 23787 df-hmeo 23980 df-fil 24071 df-fm 24163 df-flim 24164 df-flf 24165 df-xms 24545 df-ms 24546 df-tms 24547 df-cncf 25105 df-limc 26093 df-dv 26094 df-log 26789

This theorem is used by: angpieqvd 27064

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