angpieqvdlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > angpieqvdlem2		Structured version Visualization version GIF version

Theorem angpieqvdlem2 27157

Description: Equivalence used in angpieqvd 27159. (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 11669	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐵) ∈ ℂ)
4		angpieqvd.A	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ)
5	4, 2	subcld 11669	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ)
6		angpieqvd.AneB	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
7	4, 2, 6	subne0d 11679	. . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) ≠ 0)
8	3, 5, 7	divcld 12093	. . 3 ⊢ (𝜑 → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℂ)
9		angpieqvd.BneC	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
10	9	necomd 3010	. . . . 5 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
11	1, 2, 10	subne0d 11679	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐵) ≠ 0)
12	3, 5, 11, 7	divne0d 12109	. . 3 ⊢ (𝜑 → ((𝐶 − 𝐵) / (𝐴 − 𝐵)) ≠ 0)
13		lognegb 26918	. . 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 596	. 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 27132	. . 3 ⊢ (𝜑 → ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = (ℑ‘(log‘((𝐶 − 𝐵) / (𝐴 − 𝐵)))))
17	16	eqeq1d 2762	. 2 ⊢ (𝜑 → (((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π ↔ (ℑ‘(log‘((𝐶 − 𝐵) / (𝐴 − 𝐵)))) = π))
18	14, 17	bitr4d 285	1 ⊢ (𝜑 → ( -((𝐶 − 𝐵) / (𝐴 − 𝐵)) ∈ ℝ⁺ ↔ ((𝐴 − 𝐵)𝐹(𝐶 − 𝐵)) = π))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∖ cdif 4045 {csn 4583 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 ℂcc 11198 0cc0 11200 − cmin 11541 -cneg 11542 / cdiv 11973 ℝ⁺crp 13120 ℑcim 15265 πcpi 16232 logclog 26882

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-sin 16235 df-cos 16236 df-pi 16238 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187 df-log 26884

This theorem is used by: angpined 27158 angpieqvd 27159

W3C validator