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Theorem angpined 25408

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 25407	. 2 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))
8		1rp 12394	. . . . . 6 ⊢ 1 ∈ ℝ⁺
9		1re 10641	. . . . . . 7 ⊢ 1 ∈ ℝ
10		ax-1ne0 10606	. . . . . . 7 ⊢ 1 ≠ 0
11		rpneg 12422	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 1 ≠ 0) → (1 ∈ ℝ⁺ ↔ ¬ -1 ∈ ℝ⁺))
12	9, 10, 11	mp2an 690	. . . . . 6 ⊢ (1 ∈ ℝ⁺ ↔ ¬ -1 ∈ ℝ⁺)
13	8, 12	mpbi 232	. . . . 5 ⊢ ¬ -1 ∈ ℝ⁺
14	2, 3	subcld 10997	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
15	14	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴 − 𝐵) ∈ ℂ)
16	2, 3, 5	subne0d 11006	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0)
17	16	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐴 − 𝐵) ≠ 0)
18		simpr 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
19	18	oveq1d 7171	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → (𝐶 − 𝐵) = (𝐴 − 𝐵))
20	15, 17, 19	diveq1bd 11464	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐶 = 𝐴) → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) = 1)
21	20	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) = 1)
22	21	negeqd 10880	. . . . . . . 8 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -((𝐶 − 𝐵) / (𝐴 − 𝐵)) = -1)
23		simplr 767	. . . . . . . 8 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺)
24	22, 23	eqeltrrd 2914	. . . . . . 7 ⊢ (((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) ∧ 𝐶 = 𝐴) → -1 ∈ ℝ⁺)
25	24	ex 415	. . . . . 6 ⊢ ((𝜑 ∧ -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺) → (𝐶 = 𝐴 → -1 ∈ ℝ⁺))
26	25	necon3bd 3030	. . . . 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 415	. . 3 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ → 𝐶 ≠ 𝐴))
29		necom 3069	. . 3 ⊢ (𝐶 ≠ 𝐴 ↔ 𝐴 ≠ 𝐶)
30	28, 29	syl6ib 253	. 2 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ → 𝐴 ≠ 𝐶))
31	7, 30	sylbird 262	1 ⊢ (𝜑 → (((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π → 𝐴 ≠ 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 {csn 4567 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 − cmin 10870 -cneg 10871 / cdiv 11297 ℝ⁺crp 12390 ℑcim 14457 πcpi 15420 logclog 25138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140

This theorem is referenced by: angpieqvd 25409

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