isosctr - Metamath Proof Explorer

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Theorem isosctr 26903

Description: Isosceles triangle theorem. This is Metamath 100 proof #65. (Contributed by Saveliy Skresanov, 1-Jan-2017.)

**Hypothesis**
Ref	Expression
isosctrlem3.1	⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))

**Assertion**
Ref	Expression
isosctr	⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → ((𝐶 − 𝐴)𝐹(𝐵 − 𝐴)) = ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)))

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

**Proof of Theorem isosctr**
Step	Hyp	Ref	Expression
1		simp11 1203	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → 𝐴 ∈ ℂ)
2		simp13 1205	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → 𝐶 ∈ ℂ)
3	1, 2	subcld 11653	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → (𝐴 − 𝐶) ∈ ℂ)
4		simp12 1204	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → 𝐵 ∈ ℂ)
5	4, 2	subcld 11653	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → (𝐵 − 𝐶) ∈ ℂ)
6		simp21 1206	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → 𝐴 ≠ 𝐶)
7	1, 2, 6	subne0d 11662	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → (𝐴 − 𝐶) ≠ 0)
8		simp22 1207	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → 𝐵 ≠ 𝐶)
9	4, 2, 8	subne0d 11662	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → (𝐵 − 𝐶) ≠ 0)
10		simp23 1208	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → 𝐴 ≠ 𝐵)
11		subcan2 11567	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵))
12	11	3ad2ant1 1133	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → ((𝐴 − 𝐶) = (𝐵 − 𝐶) ↔ 𝐴 = 𝐵))
13	12	necon3bid 2991	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → ((𝐴 − 𝐶) ≠ (𝐵 − 𝐶) ↔ 𝐴 ≠ 𝐵))
14	10, 13	mpbird 257	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → (𝐴 − 𝐶) ≠ (𝐵 − 𝐶))
15		simp3 1138	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶)))
16		isosctrlem3.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
17	16	isosctrlem3 26902	. . 3 ⊢ ((((𝐴 − 𝐶) ∈ ℂ ∧ (𝐵 − 𝐶) ∈ ℂ) ∧ ((𝐴 − 𝐶) ≠ 0 ∧ (𝐵 − 𝐶) ≠ 0 ∧ (𝐴 − 𝐶) ≠ (𝐵 − 𝐶)) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → (-(𝐴 − 𝐶)𝐹((𝐵 − 𝐶) − (𝐴 − 𝐶))) = (((𝐴 − 𝐶) − (𝐵 − 𝐶))𝐹-(𝐵 − 𝐶)))
18	3, 5, 7, 9, 14, 15, 17	syl231anc 1390	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → (-(𝐴 − 𝐶)𝐹((𝐵 − 𝐶) − (𝐴 − 𝐶))) = (((𝐴 − 𝐶) − (𝐵 − 𝐶))𝐹-(𝐵 − 𝐶)))
19	1, 2	negsubdi2d 11669	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → -(𝐴 − 𝐶) = (𝐶 − 𝐴))
20	4, 1, 2	nnncan2d 11688	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → ((𝐵 − 𝐶) − (𝐴 − 𝐶)) = (𝐵 − 𝐴))
21	19, 20	oveq12d 7470	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → (-(𝐴 − 𝐶)𝐹((𝐵 − 𝐶) − (𝐴 − 𝐶))) = ((𝐶 − 𝐴)𝐹(𝐵 − 𝐴)))
22	1, 4, 2	nnncan2d 11688	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → ((𝐴 − 𝐶) − (𝐵 − 𝐶)) = (𝐴 − 𝐵))
23	4, 2	negsubdi2d 11669	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → -(𝐵 − 𝐶) = (𝐶 − 𝐵))
24	22, 23	oveq12d 7470	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → (((𝐴 − 𝐶) − (𝐵 − 𝐶))𝐹-(𝐵 − 𝐶)) = ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)))
25	18, 21, 24	3eqtr3d 2788	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ 𝐴 ≠ 𝐵) ∧ (abs‘(𝐴 − 𝐶)) = (abs‘(𝐵 − 𝐶))) → ((𝐶 − 𝐴)𝐹(𝐵 − 𝐴)) = ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∖ cdif 3974 {csn 4649 ‘cfv 6577 (class class class)co 7452 ∈ cmpo 7454 ℂcc 11186 0cc0 11188 − cmin 11525 -cneg 11526 / cdiv 11953 ℑcim 15165 abscabs 15301 logclog 26635

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5305 ax-sep 5319 ax-nul 5326 ax-pow 5385 ax-pr 5449 ax-un 7774 ax-inf2 9714 ax-cnex 11244 ax-resscn 11245 ax-1cn 11246 ax-icn 11247 ax-addcl 11248 ax-addrcl 11249 ax-mulcl 11250 ax-mulrcl 11251 ax-mulcom 11252 ax-addass 11253 ax-mulass 11254 ax-distr 11255 ax-i2m1 11256 ax-1ne0 11257 ax-1rid 11258 ax-rnegex 11259 ax-rrecex 11260 ax-cnre 11261 ax-pre-lttri 11262 ax-pre-lttrn 11263 ax-pre-ltadd 11264 ax-pre-mulgt0 11265 ax-pre-sup 11266 ax-addf 11267

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3445 df-v 3491 df-sbc 3806 df-csb 3923 df-dif 3980 df-un 3982 df-in 3984 df-ss 3994 df-pss 3997 df-nul 4354 df-if 4550 df-pw 4625 df-sn 4650 df-pr 4652 df-tp 4654 df-op 4656 df-uni 4934 df-int 4973 df-iun 5019 df-iin 5020 df-br 5169 df-opab 5231 df-mpt 5252 df-tr 5286 df-id 5595 df-eprel 5601 df-po 5609 df-so 5610 df-fr 5654 df-se 5655 df-we 5656 df-xp 5708 df-rel 5709 df-cnv 5710 df-co 5711 df-dm 5712 df-rn 5713 df-res 5714 df-ima 5715 df-pred 6336 df-ord 6402 df-on 6403 df-lim 6404 df-suc 6405 df-iota 6529 df-fun 6579 df-fn 6580 df-f 6581 df-f1 6582 df-fo 6583 df-f1o 6584 df-fv 6585 df-isom 6586 df-riota 7408 df-ov 7455 df-oprab 7456 df-mpo 7457 df-of 7718 df-om 7908 df-1st 8034 df-2nd 8035 df-supp 8206 df-frecs 8326 df-wrecs 8357 df-recs 8431 df-rdg 8470 df-1o 8526 df-2o 8527 df-er 8767 df-map 8890 df-pm 8891 df-ixp 8960 df-en 9008 df-dom 9009 df-sdom 9010 df-fin 9011 df-fsupp 9436 df-fi 9484 df-sup 9515 df-inf 9516 df-oi 9583 df-card 10012 df-pnf 11330 df-mnf 11331 df-xr 11332 df-ltxr 11333 df-le 11334 df-sub 11527 df-neg 11528 df-div 11954 df-nn 12300 df-2 12362 df-3 12363 df-4 12364 df-5 12365 df-6 12366 df-7 12367 df-8 12368 df-9 12369 df-n0 12560 df-z 12647 df-dec 12767 df-uz 12912 df-q 13024 df-rp 13068 df-xneg 13187 df-xadd 13188 df-xmul 13189 df-ioo 13423 df-ioc 13424 df-ico 13425 df-icc 13426 df-fz 13580 df-fzo 13724 df-fl 13860 df-mod 13938 df-seq 14071 df-exp 14131 df-fac 14341 df-bc 14370 df-hash 14398 df-shft 15134 df-cj 15166 df-re 15167 df-im 15168 df-sqrt 15302 df-abs 15303 df-limsup 15535 df-clim 15552 df-rlim 15553 df-sum 15753 df-ef 16133 df-sin 16135 df-cos 16136 df-pi 16138 df-struct 17215 df-sets 17232 df-slot 17250 df-ndx 17262 df-base 17280 df-ress 17309 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17503 df-topn 17504 df-0g 17522 df-gsum 17523 df-topgen 17524 df-pt 17525 df-prds 17528 df-xrs 17583 df-qtop 17588 df-imas 17589 df-xps 17591 df-mre 17665 df-mrc 17666 df-acs 17668 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18840 df-mulg 19129 df-cntz 19378 df-cmn 19845 df-psmet 21400 df-xmet 21401 df-met 21402 df-bl 21403 df-mopn 21404 df-fbas 21405 df-fg 21406 df-cnfld 21409 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22995 df-cld 23069 df-ntr 23070 df-cls 23071 df-nei 23148 df-lp 23186 df-perf 23187 df-cn 23277 df-cnp 23278 df-haus 23365 df-tx 23612 df-hmeo 23805 df-fil 23896 df-fm 23988 df-flim 23989 df-flf 23990 df-xms 24372 df-ms 24373 df-tms 24374 df-cncf 24944 df-limc 25942 df-dv 25943 df-log 26637

This theorem is referenced by: (None)

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