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Theorem footeq 28556

Description: Uniqueness of the foot point. (Contributed by Thierry Arnoux, 1-Mar-2020.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
footeq	⊢ (𝜑 → 𝑋 = 𝑌)

Proof of Theorem footeq Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7434	. . 3 ⊢ (𝑥 = 𝑋 → (𝑍𝐿𝑥) = (𝑍𝐿𝑋))
2	1	breq1d 5162	. 2 ⊢ (𝑥 = 𝑋 → ((𝑍𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝑍𝐿𝑋)(⟂G‘𝐺)𝐴))
3		oveq2 7434	. . 3 ⊢ (𝑥 = 𝑌 → (𝑍𝐿𝑥) = (𝑍𝐿𝑌))
4	3	breq1d 5162	. 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 28555	. . 3 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝐴)
15	5, 6, 7, 8, 9, 10, 11, 14	foot 28554	. 2 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (𝑍𝐿𝑥)(⟂G‘𝐺)𝐴)
16		footeq.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
17	5, 8, 7, 9, 10, 12	tglnpt 28381	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
18	8, 9, 13	perpln1 28542	. . . . 5 ⊢ (𝜑 → (𝑋𝐿𝑍) ∈ ran 𝐿)
19	5, 7, 8, 9, 17, 11, 18	tglnne 28460	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
20	5, 7, 8, 9, 17, 11, 19	tglinecom 28467	. . 3 ⊢ (𝜑 → (𝑋𝐿𝑍) = (𝑍𝐿𝑋))
21	20, 13	eqbrtrrd 5176	. 2 ⊢ (𝜑 → (𝑍𝐿𝑋)(⟂G‘𝐺)𝐴)
22	5, 8, 7, 9, 10, 16	tglnpt 28381	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
23		footeq.2	. . . . . 6 ⊢ (𝜑 → (𝑌𝐿𝑍)(⟂G‘𝐺)𝐴)
24	8, 9, 23	perpln1 28542	. . . . 5 ⊢ (𝜑 → (𝑌𝐿𝑍) ∈ ran 𝐿)
25	5, 7, 8, 9, 22, 11, 24	tglnne 28460	. . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
26	5, 7, 8, 9, 22, 11, 25	tglinecom 28467	. . 3 ⊢ (𝜑 → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
27	26, 23	eqbrtrrd 5176	. 2 ⊢ (𝜑 → (𝑍𝐿𝑌)(⟂G‘𝐺)𝐴)
28	2, 4, 15, 12, 16, 21, 27	reu2eqd 3733	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ran crn 5683 ‘cfv 6553 (class class class)co 7426 Basecbs 17189 distcds 17251 TarskiGcstrkg 28259 Itvcitv 28265 LineGclng 28266 ⟂Gcperpg 28527

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 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 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-oadd 8499 df-er 8733 df-map 8855 df-pm 8856 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-dju 9934 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-xnn0 12585 df-z 12599 df-uz 12863 df-fz 13527 df-fzo 13670 df-hash 14332 df-word 14507 df-concat 14563 df-s1 14588 df-s2 14841 df-s3 14842 df-trkgc 28280 df-trkgb 28281 df-trkgcb 28282 df-trkg 28285 df-cgrg 28343 df-leg 28415 df-mir 28485 df-rag 28526 df-perpg 28528

This theorem is referenced by: (None)

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