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Theorem angpieqvdlem2 25123

Description: Equivalence used in angpieqvd 25125. (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
angpieqvdlem2	⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))

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

**Proof of Theorem angpieqvdlem2**
Step	Hyp	Ref	Expression
1		angpieqvd.C	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ)
2		angpieqvd.B	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3	1, 2	subcld 10796	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ)
4		angpieqvd.A	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
5	4, 2	subcld 10796	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
6		angpieqvd.AneB	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
7	4, 2, 6	subne0d 10805	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0)
8	3, 5, 7	divcld 11215	. . 3 ⊢ (𝜑 → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℂ)
9		angpieqvd.BneC	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
10	9	necomd 3015	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
11	1, 2, 10	subne0d 10805	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐵) ≠ 0)
12	3, 5, 11, 7	divne0d 11231	. . 3 ⊢ (𝜑 → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) ≠ 0)
13		lognegb 24889	. . 3 ⊢ ((((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℂ ∧ ((𝐶 − 𝐵) / (𝐴 − 𝐵)) ≠ 0) → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ (ℑ‘(log‘((𝐶 − 𝐵) / (𝐴 − 𝐵)))) = π))
14	8, 12, 13	syl2anc 576	. 2 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ (ℑ‘(log‘((𝐶 − 𝐵) / (𝐴 − 𝐵)))) = π))
15		angpieqvd.angdef	. . . 4 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
16	15, 5, 7, 3, 11	angvald 25098	. . 3 ⊢ (𝜑 → ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = (ℑ‘(log‘((𝐶 − 𝐵) / (𝐴 − 𝐵)))))
17	16	eqeq1d 2773	. 2 ⊢ (𝜑 → (((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π ↔ (ℑ‘(log‘((𝐶 − 𝐵) / (𝐴 − 𝐵)))) = π))
18	14, 17	bitr4d 274	1 ⊢ (𝜑 → (-((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 ∖ cdif 3819 {csn 4435 ‘cfv 6185 (class class class)co 6974 ∈ cmpo 6976 ℂcc 10331 0cc0 10333 − cmin 10668 -cneg 10669 / cdiv 11096 ℝ⁺crp 12202 ℑcim 14316 πcpi 15278 logclog 24854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-cda 9386 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-ioc 12557 df-ico 12558 df-icc 12559 df-fz 12707 df-fzo 12848 df-fl 12975 df-mod 13051 df-seq 13183 df-exp 13243 df-fac 13447 df-bc 13476 df-hash 13504 df-shft 14285 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-limsup 14687 df-clim 14704 df-rlim 14705 df-sum 14902 df-ef 15279 df-sin 15281 df-cos 15282 df-pi 15284 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-pt 16572 df-prds 16575 df-xrs 16629 df-qtop 16634 df-imas 16635 df-xps 16637 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-fbas 20259 df-fg 20260 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-ntr 21347 df-cls 21348 df-nei 21425 df-lp 21463 df-perf 21464 df-cn 21554 df-cnp 21555 df-haus 21642 df-tx 21889 df-hmeo 22082 df-fil 22173 df-fm 22265 df-flim 22266 df-flf 22267 df-xms 22648 df-ms 22649 df-tms 22650 df-cncf 23204 df-limc 24182 df-dv 24183 df-log 24856

This theorem is referenced by: angpined 25124 angpieqvd 25125

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