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Mirrors > Home > MPE Home > Th. List > angrtmuld | Structured version Visualization version GIF version |
Description: Perpendicularity of two vectors does not change under rescaling the second. (Contributed by David Moews, 28-Feb-2017.) |
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) |
Ref | Expression |
---|---|
angrtmuld | ⊢ (𝜑 → ((𝑋𝐹𝑌) ∈ {(π / 2), -(π / 2)} ↔ (𝑋𝐹𝑍) ∈ {(π / 2), -(π / 2)})) |
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 11167 | . . . 4 ⊢ (𝜑 → (𝑍 / 𝑌) ≠ 0) |
6 | 5 | neneqd 2974 | . . 3 ⊢ (𝜑 → ¬ (𝑍 / 𝑌) = 0) |
7 | biorf 923 | . . 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 24984 | . 2 ⊢ (𝜑 → ((𝑋𝐹𝑌) ∈ {(π / 2), -(π / 2)} ↔ (ℜ‘(𝑌 / 𝑋)) = 0)) |
13 | 9, 10, 11, 1, 3 | angrteqvd 24984 | . . 3 ⊢ (𝜑 → ((𝑋𝐹𝑍) ∈ {(π / 2), -(π / 2)} ↔ (ℜ‘(𝑍 / 𝑋)) = 0)) |
14 | 1, 2, 10, 4, 11 | dmdcan2d 11181 | . . . . . 6 ⊢ (𝜑 → ((𝑍 / 𝑌) · (𝑌 / 𝑋)) = (𝑍 / 𝑋)) |
15 | 14 | fveq2d 6450 | . . . . 5 ⊢ (𝜑 → (ℜ‘((𝑍 / 𝑌) · (𝑌 / 𝑋))) = (ℜ‘(𝑍 / 𝑋))) |
16 | angrtmuld.7 | . . . . . 6 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℝ) | |
17 | 2, 10, 11 | divcld 11151 | . . . . . 6 ⊢ (𝜑 → (𝑌 / 𝑋) ∈ ℂ) |
18 | 16, 17 | remul2d 14374 | . . . . 5 ⊢ (𝜑 → (ℜ‘((𝑍 / 𝑌) · (𝑌 / 𝑋))) = ((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋)))) |
19 | 15, 18 | eqtr3d 2816 | . . . 4 ⊢ (𝜑 → (ℜ‘(𝑍 / 𝑋)) = ((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋)))) |
20 | 19 | eqeq1d 2780 | . . 3 ⊢ (𝜑 → ((ℜ‘(𝑍 / 𝑋)) = 0 ↔ ((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋))) = 0)) |
21 | 1, 2, 4 | divcld 11151 | . . . 4 ⊢ (𝜑 → (𝑍 / 𝑌) ∈ ℂ) |
22 | 17 | recld 14341 | . . . . 5 ⊢ (𝜑 → (ℜ‘(𝑌 / 𝑋)) ∈ ℝ) |
23 | 22 | recnd 10405 | . . . 4 ⊢ (𝜑 → (ℜ‘(𝑌 / 𝑋)) ∈ ℂ) |
24 | 21, 23 | mul0ord 11025 | . . 3 ⊢ (𝜑 → (((𝑍 / 𝑌) · (ℜ‘(𝑌 / 𝑋))) = 0 ↔ ((𝑍 / 𝑌) = 0 ∨ (ℜ‘(𝑌 / 𝑋)) = 0))) |
25 | 13, 20, 24 | 3bitrd 297 | . 2 ⊢ (𝜑 → ((𝑋𝐹𝑍) ∈ {(π / 2), -(π / 2)} ↔ ((𝑍 / 𝑌) = 0 ∨ (ℜ‘(𝑌 / 𝑋)) = 0))) |
26 | 8, 12, 25 | 3bitr4d 303 | 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 198 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∖ cdif 3789 {csn 4398 {cpr 4400 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 ℂcc 10270 ℝcr 10271 0cc0 10272 · cmul 10277 -cneg 10607 / cdiv 11032 2c2 11430 ℜcre 14244 ℑcim 14245 πcpi 15199 logclog 24738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-ef 15200 df-sin 15202 df-cos 15203 df-pi 15205 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cncf 23089 df-limc 24067 df-dv 24068 df-log 24740 |
This theorem is referenced by: chordthmlem2 25011 |
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