angrtmuld - Metamath Proof Explorer

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Theorem angrtmuld 26699

Description: Perpendicularity of two vectors does not change under rescaling the second. (Contributed by David Moews, 28-Feb-2017.)

**Hypotheses**
Ref	Expression
ang.1	⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
angrtmuld.1	⊢ (𝜑 → 𝑋 ∈ ℂ)
angrtmuld.2	⊢ (𝜑 → 𝑌 ∈ ℂ)
angrtmuld.3	⊢ (𝜑 → 𝑍 ∈ ℂ)
angrtmuld.4	⊢ (𝜑 → 𝑋 ≠ 0)
angrtmuld.5	⊢ (𝜑 → 𝑌 ≠ 0)
angrtmuld.6	⊢ (𝜑 → 𝑍 ≠ 0)
angrtmuld.7	⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ)

**Assertion**
Ref	Expression
angrtmuld	⊢ (𝜑 → ((𝑋𝐹𝑌) ∈ {(π / 2), -(π / 2)} ↔ (𝑋𝐹𝑍) ∈ {(π / 2), -(π / 2)}))

Distinct variable groups: 𝑥,𝑦,𝑋 𝑥,𝑌,𝑦 𝑥,𝑍,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐹(𝑥,𝑦)

**Proof of Theorem angrtmuld**
Step	Hyp	Ref	Expression
1		angrtmuld.3	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ ℂ)
2		angrtmuld.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℂ)
3		angrtmuld.6	. . . . 5 ⊢ (𝜑 → 𝑍 ≠ 0)
4		angrtmuld.5	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0)
5	1, 2, 3, 4	divne0d 11904	. . . 4 ⊢ (𝜑 → (𝑍 / 𝑌) ≠ 0)
6	5	neneqd 2930	. . 3 ⊢ (𝜑 → ¬ (𝑍 / 𝑌) = 0)
7		biorf 936	. . 3 ⊢ (¬ (𝑍 / 𝑌) = 0 → ((ℜ‘(𝑌 / 𝑋)) = 0 ↔ ((𝑍 / 𝑌) = 0 ∨ (ℜ‘(𝑌 / 𝑋)) = 0)))
8	6, 7	syl 17	. 2 ⊢ (𝜑 → ((ℜ‘(𝑌 / 𝑋)) = 0 ↔ ((𝑍 / 𝑌) = 0 ∨ (ℜ‘(𝑌 / 𝑋)) = 0)))
9		ang.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (ℑ‘(log‘(𝑦 / 𝑥))))
10		angrtmuld.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ)
11		angrtmuld.4	. . 3 ⊢ (𝜑 → 𝑋 ≠ 0)
12	9, 10, 11, 2, 4	angrteqvd 26697	. 2 ⊢ (𝜑 → ((𝑋𝐹𝑌) ∈ {(π / 2), -(π / 2)} ↔ (ℜ‘(𝑌 / 𝑋)) = 0))
13	9, 10, 11, 1, 3	angrteqvd 26697	. . 3 ⊢ (𝜑 → ((𝑋𝐹𝑍) ∈ {(π / 2), -(π / 2)} ↔ (ℜ‘(𝑍 / 𝑋)) = 0))
14	1, 2, 10, 4, 11	dmdcan2d 11918	. . . . . 6 ⊢ (𝜑 → ((𝑍 / 𝑌) · (𝑌 / 𝑋)) = (𝑍 / 𝑋))
15	14	fveq2d 6820	. . . . 5 ⊢ (𝜑 → (ℜ‘((𝑍 / 𝑌) · (𝑌 / 𝑋))) = (ℜ‘(𝑍 / 𝑋)))
16		angrtmuld.7	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
17	2, 10, 11	divcld 11888	. . . . . 6 ⊢ (𝜑 → (𝑌 / 𝑋) ∈ ℂ)
18	16, 17	remul2d 15121	. . . . 5 ⊢ (𝜑 → (ℜ‘((𝑍 / 𝑌) · (𝑌 / 𝑋))) = ((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋))))
19	15, 18	eqtr3d 2766	. . . 4 ⊢ (𝜑 → (ℜ‘(𝑍 / 𝑋)) = ((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋))))
20	19	eqeq1d 2731	. . 3 ⊢ (𝜑 → ((ℜ‘(𝑍 / 𝑋)) = 0 ↔ ((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋))) = 0))
21	1, 2, 4	divcld 11888	. . . 4 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℂ)
22	17	recld 15088	. . . . 5 ⊢ (𝜑 → (ℜ‘(𝑌 / 𝑋)) ∈ ℝ)
23	22	recnd 11131	. . . 4 ⊢ (𝜑 → (ℜ‘(𝑌 / 𝑋)) ∈ ℂ)
24	21, 23	mul0ord 11756	. . 3 ⊢ (𝜑 → (((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋))) = 0 ↔ ((𝑍 / 𝑌) = 0 ∨ (ℜ‘(𝑌 / 𝑋)) = 0)))
25	13, 20, 24	3bitrd 305	. 2 ⊢ (𝜑 → ((𝑋𝐹𝑍) ∈ {(π / 2), -(π / 2)} ↔ ((𝑍 / 𝑌) = 0 ∨ (ℜ‘(𝑌 / 𝑋)) = 0)))
26	8, 12, 25	3bitr4d 311	1 ⊢ (𝜑 → ((𝑋𝐹𝑌) ∈ {(π / 2), -(π / 2)} ↔ (𝑋𝐹𝑍) ∈ {(π / 2), -(π / 2)}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3896 {csn 4573 {cpr 4575 ‘cfv 6476 (class class class)co 7340 ∈ cmpo 7342 ℂcc 10995 ℝcr 10996 0cc0 10997 · cmul 11002 -cneg 11336 / cdiv 11765 2c2 12171 ℜcre 14991 ℑcim 14992 πcpi 15960 logclog 26444

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5214 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-inf2 9525 ax-cnex 11053 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073 ax-pre-mulgt0 11074 ax-pre-sup 11075 ax-addf 11076

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4895 df-iun 4940 df-iin 4941 df-br 5089 df-opab 5151 df-mpt 5170 df-tr 5196 df-id 5508 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5566 df-se 5567 df-we 5568 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7297 df-ov 7343 df-oprab 7344 df-mpo 7345 df-of 7604 df-om 7791 df-1st 7915 df-2nd 7916 df-supp 8085 df-frecs 8205 df-wrecs 8236 df-recs 8285 df-rdg 8323 df-1o 8379 df-2o 8380 df-er 8616 df-map 8746 df-pm 8747 df-ixp 8816 df-en 8864 df-dom 8865 df-sdom 8866 df-fin 8867 df-fsupp 9240 df-fi 9289 df-sup 9320 df-inf 9321 df-oi 9390 df-card 9823 df-pnf 11139 df-mnf 11140 df-xr 11141 df-ltxr 11142 df-le 11143 df-sub 11337 df-neg 11338 df-div 11766 df-nn 12117 df-2 12179 df-3 12180 df-4 12181 df-5 12182 df-6 12183 df-7 12184 df-8 12185 df-9 12186 df-n0 12373 df-z 12460 df-dec 12580 df-uz 12724 df-q 12838 df-rp 12882 df-xneg 13002 df-xadd 13003 df-xmul 13004 df-ioo 13240 df-ioc 13241 df-ico 13242 df-icc 13243 df-fz 13399 df-fzo 13546 df-fl 13684 df-mod 13762 df-seq 13897 df-exp 13957 df-fac 14169 df-bc 14198 df-hash 14226 df-shft 14961 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-limsup 15365 df-clim 15382 df-rlim 15383 df-sum 15581 df-ef 15961 df-sin 15963 df-cos 15964 df-pi 15966 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17108 df-ress 17129 df-plusg 17161 df-mulr 17162 df-starv 17163 df-sca 17164 df-vsca 17165 df-ip 17166 df-tset 17167 df-ple 17168 df-ds 17170 df-unif 17171 df-hom 17172 df-cco 17173 df-rest 17313 df-topn 17314 df-0g 17332 df-gsum 17333 df-topgen 17334 df-pt 17335 df-prds 17338 df-xrs 17393 df-qtop 17398 df-imas 17399 df-xps 17401 df-mre 17475 df-mrc 17476 df-acs 17478 df-mgm 18501 df-sgrp 18580 df-mnd 18596 df-submnd 18645 df-mulg 18934 df-cntz 19183 df-cmn 19648 df-psmet 21237 df-xmet 21238 df-met 21239 df-bl 21240 df-mopn 21241 df-fbas 21242 df-fg 21243 df-cnfld 21246 df-top 22763 df-topon 22780 df-topsp 22802 df-bases 22815 df-cld 22888 df-ntr 22889 df-cls 22890 df-nei 22967 df-lp 23005 df-perf 23006 df-cn 23096 df-cnp 23097 df-haus 23184 df-tx 23431 df-hmeo 23624 df-fil 23715 df-fm 23807 df-flim 23808 df-flf 23809 df-xms 24189 df-ms 24190 df-tms 24191 df-cncf 24752 df-limc 25748 df-dv 25749 df-log 26446

This theorem is referenced by: chordthmlem2 26724

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