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Mirrors > Home > MPE Home > Th. List > footeq | Structured version Visualization version GIF version |
Description: Uniqueness of the foot point. (Contributed by Thierry Arnoux, 1-Mar-2020.) |
Ref | Expression |
---|---|
isperp.p | ⊢ 𝑃 = (Base‘𝐺) |
isperp.d | ⊢ − = (dist‘𝐺) |
isperp.i | ⊢ 𝐼 = (Itv‘𝐺) |
isperp.l | ⊢ 𝐿 = (LineG‘𝐺) |
isperp.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
isperp.a | ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) |
footeq.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
footeq.y | ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
footeq.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
footeq.1 | ⊢ (𝜑 → (𝑋𝐿𝑍)(⟂G‘𝐺)𝐴) |
footeq.2 | ⊢ (𝜑 → (𝑌𝐿𝑍)(⟂G‘𝐺)𝐴) |
Ref | Expression |
---|---|
footeq | ⊢ (𝜑 → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6984 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑍𝐿𝑥) = (𝑍𝐿𝑋)) | |
2 | 1 | breq1d 4939 | . 2 ⊢ (𝑥 = 𝑋 → ((𝑍𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝑍𝐿𝑋)(⟂G‘𝐺)𝐴)) |
3 | oveq2 6984 | . . 3 ⊢ (𝑥 = 𝑌 → (𝑍𝐿𝑥) = (𝑍𝐿𝑌)) | |
4 | 3 | breq1d 4939 | . 2 ⊢ (𝑥 = 𝑌 → ((𝑍𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝑍𝐿𝑌)(⟂G‘𝐺)𝐴)) |
5 | isperp.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
6 | isperp.d | . . 3 ⊢ − = (dist‘𝐺) | |
7 | isperp.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
8 | isperp.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
9 | isperp.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
10 | isperp.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) | |
11 | footeq.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
12 | footeq.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
13 | footeq.1 | . . . 4 ⊢ (𝜑 → (𝑋𝐿𝑍)(⟂G‘𝐺)𝐴) | |
14 | 5, 6, 7, 8, 9, 10, 12, 11, 13 | footne 26211 | . . 3 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝐴) |
15 | 5, 6, 7, 8, 9, 10, 11, 14 | foot 26210 | . 2 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (𝑍𝐿𝑥)(⟂G‘𝐺)𝐴) |
16 | footeq.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | |
17 | 5, 8, 7, 9, 10, 12 | tglnpt 26037 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
18 | 8, 9, 13 | perpln1 26198 | . . . . 5 ⊢ (𝜑 → (𝑋𝐿𝑍) ∈ ran 𝐿) |
19 | 5, 7, 8, 9, 17, 11, 18 | tglnne 26116 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑍) |
20 | 5, 7, 8, 9, 17, 11, 19 | tglinecom 26123 | . . 3 ⊢ (𝜑 → (𝑋𝐿𝑍) = (𝑍𝐿𝑋)) |
21 | 20, 13 | eqbrtrrd 4953 | . 2 ⊢ (𝜑 → (𝑍𝐿𝑋)(⟂G‘𝐺)𝐴) |
22 | 5, 8, 7, 9, 10, 16 | tglnpt 26037 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
23 | footeq.2 | . . . . . 6 ⊢ (𝜑 → (𝑌𝐿𝑍)(⟂G‘𝐺)𝐴) | |
24 | 8, 9, 23 | perpln1 26198 | . . . . 5 ⊢ (𝜑 → (𝑌𝐿𝑍) ∈ ran 𝐿) |
25 | 5, 7, 8, 9, 22, 11, 24 | tglnne 26116 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑍) |
26 | 5, 7, 8, 9, 22, 11, 25 | tglinecom 26123 | . . 3 ⊢ (𝜑 → (𝑌𝐿𝑍) = (𝑍𝐿𝑌)) |
27 | 26, 23 | eqbrtrrd 4953 | . 2 ⊢ (𝜑 → (𝑍𝐿𝑌)(⟂G‘𝐺)𝐴) |
28 | 2, 4, 15, 12, 16, 21, 27 | reu2eqd 3639 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 class class class wbr 4929 ran crn 5408 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 distcds 16430 TarskiGcstrkg 25918 Itvcitv 25924 LineGclng 25925 ⟂Gcperpg 26183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-dju 9124 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-n0 11708 df-xnn0 11780 df-z 11794 df-uz 12059 df-fz 12709 df-fzo 12850 df-hash 13506 df-word 13673 df-concat 13734 df-s1 13759 df-s2 14072 df-s3 14073 df-trkgc 25936 df-trkgb 25937 df-trkgcb 25938 df-trkg 25941 df-cgrg 25999 df-leg 26071 df-mir 26141 df-rag 26182 df-perpg 26184 |
This theorem is referenced by: (None) |
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