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Mirrors > Home > MPE Home > Th. List > footne | Structured version Visualization version GIF version |
Description: Uniqueness of the foot point. (Contributed by Thierry Arnoux, 28-Feb-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) |
footne.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
footne.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
footne.1 | ⊢ (𝜑 → (𝑋𝐿𝑌)(⟂G‘𝐺)𝐴) |
Ref | Expression |
---|---|
footne | ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isperp.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | isperp.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | isperp.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | isperp.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝐺 ∈ TarskiG) |
6 | isperp.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) | |
7 | 6 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝐴 ∈ ran 𝐿) |
8 | footne.1 | . . . . . 6 ⊢ (𝜑 → (𝑋𝐿𝑌)(⟂G‘𝐺)𝐴) | |
9 | 3, 4, 8 | perpln1 28637 | . . . . 5 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) |
10 | 9 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (𝑋𝐿𝑌) ∈ ran 𝐿) |
11 | isperp.d | . . . . . . 7 ⊢ − = (dist‘𝐺) | |
12 | 1, 11, 2, 3, 4, 9, 6, 8 | perpneq 28641 | . . . . . 6 ⊢ (𝜑 → (𝑋𝐿𝑌) ≠ 𝐴) |
13 | 12 | necomd 2986 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ (𝑋𝐿𝑌)) |
14 | 13 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝐴 ≠ (𝑋𝐿𝑌)) |
15 | footne.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
16 | 15 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐴) |
17 | 1, 3, 2, 4, 6, 15 | tglnpt 28476 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
18 | footne.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
19 | 1, 2, 3, 4, 17, 18, 9 | tglnne 28555 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
20 | 1, 2, 3, 4, 17, 18, 19 | tglinerflx1 28560 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑋𝐿𝑌)) |
21 | 20 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ (𝑋𝐿𝑌)) |
22 | 16, 21 | elind 4195 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ (𝐴 ∩ (𝑋𝐿𝑌))) |
23 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐴) | |
24 | 1, 2, 3, 4, 17, 18, 19 | tglinerflx2 28561 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐿𝑌)) |
25 | 24 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ (𝑋𝐿𝑌)) |
26 | 23, 25 | elind 4195 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ (𝐴 ∩ (𝑋𝐿𝑌))) |
27 | 1, 2, 3, 5, 7, 10, 14, 22, 26 | tglineineq 28570 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝑋 = 𝑌) |
28 | 19 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → 𝑋 ≠ 𝑌) |
29 | 27, 28 | pm2.21ddne 3016 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → ¬ 𝑌 ∈ 𝐴) |
30 | 29 | pm2.01da 797 | 1 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 class class class wbr 5153 ran crn 5683 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 distcds 17275 TarskiGcstrkg 28354 Itvcitv 28360 LineGclng 28361 ⟂Gcperpg 28622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-oadd 8500 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12597 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-hash 14348 df-word 14523 df-concat 14579 df-s1 14604 df-s2 14857 df-s3 14858 df-trkgc 28375 df-trkgb 28376 df-trkgcb 28377 df-trkg 28380 df-cgrg 28438 df-mir 28580 df-rag 28621 df-perpg 28623 |
This theorem is referenced by: footeq 28651 hlperpnel 28652 oppperpex 28680 |
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