angrtmuld - Metamath Proof Explorer

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Theorem angrtmuld 26774

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 11933	. . . 4 ⊢ (𝜑 → (𝑍 / 𝑌) ≠ 0)
6	5	neneqd 2937	. . 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 26772	. 2 ⊢ (𝜑 → ((𝑋𝐹𝑌) ∈ {(π / 2), -(π / 2)} ↔ (ℜ‘(𝑌 / 𝑋)) = 0))
13	9, 10, 11, 1, 3	angrteqvd 26772	. . 3 ⊢ (𝜑 → ((𝑋𝐹𝑍) ∈ {(π / 2), -(π / 2)} ↔ (ℜ‘(𝑍 / 𝑋)) = 0))
14	1, 2, 10, 4, 11	dmdcan2d 11947	. . . . . 6 ⊢ (𝜑 → ((𝑍 / 𝑌) · (𝑌 / 𝑋)) = (𝑍 / 𝑋))
15	14	fveq2d 6838	. . . . 5 ⊢ (𝜑 → (ℜ‘((𝑍 / 𝑌) · (𝑌 / 𝑋))) = (ℜ‘(𝑍 / 𝑋)))
16		angrtmuld.7	. . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ)
17	2, 10, 11	divcld 11917	. . . . . 6 ⊢ (𝜑 → (𝑌 / 𝑋) ∈ ℂ)
18	16, 17	remul2d 15150	. . . . 5 ⊢ (𝜑 → (ℜ‘((𝑍 / 𝑌) · (𝑌 / 𝑋))) = ((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋))))
19	15, 18	eqtr3d 2773	. . . 4 ⊢ (𝜑 → (ℜ‘(𝑍 / 𝑋)) = ((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋))))
20	19	eqeq1d 2738	. . 3 ⊢ (𝜑 → ((ℜ‘(𝑍 / 𝑋)) = 0 ↔ ((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋))) = 0))
21	1, 2, 4	divcld 11917	. . . 4 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℂ)
22	17	recld 15117	. . . . 5 ⊢ (𝜑 → (ℜ‘(𝑌 / 𝑋)) ∈ ℝ)
23	22	recnd 11160	. . . 4 ⊢ (𝜑 → (ℜ‘(𝑌 / 𝑋)) ∈ ℂ)
24	21, 23	mul0ord 11785	. . 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 1541 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 {csn 4580 {cpr 4582 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 ℂcc 11024 ℝcr 11025 0cc0 11026 · cmul 11031 -cneg 11365 / cdiv 11794 2c2 12200 ℜcre 15020 ℑcim 15021 πcpi 15989 logclog 26519

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-pi 15995 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824 df-log 26521

This theorem is referenced by: chordthmlem2 26799

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