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Theorem hlcgreulem 28601

Description: Lemma for hlcgreu 28602. (Contributed by Thierry Arnoux, 9-Aug-2020.)

**Hypotheses**
Ref	Expression
ishlg.p	⊢ 𝑃 = (Base‘𝐺)
ishlg.i	⊢ 𝐼 = (Itv‘𝐺)
ishlg.k	⊢ 𝐾 = (hlG‘𝐺)
ishlg.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
ishlg.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
ishlg.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
hlln.1	⊢ (𝜑 → 𝐺 ∈ TarskiG)
hltr.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
hlcgrex.m	⊢ − = (dist‘𝐺)
hlcgrex.1	⊢ (𝜑 → 𝐷 ≠ 𝐴)
hlcgrex.2	⊢ (𝜑 → 𝐵 ≠ 𝐶)
hlcgreulem.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
hlcgreulem.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
hlcgreulem.1	⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝐷)
hlcgreulem.2	⊢ (𝜑 → 𝑌(𝐾‘𝐴)𝐷)
hlcgreulem.3	⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶))
hlcgreulem.4	⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶))

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

Proof of Theorem hlcgreulem Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ishlg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		hlcgrex.m	. . 3 ⊢ − = (dist‘𝐺)
3		ishlg.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		hlln.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐺 ∈ TarskiG)
6		ishlg.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ 𝑃)
8		ishlg.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐵 ∈ 𝑃)
10		ishlg.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐶 ∈ 𝑃)
12		simplr 768	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ∈ 𝑃)
13		hlcgreulem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
14	13	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 ∈ 𝑃)
15		hlcgreulem.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
16	15	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑌 ∈ 𝑃)
17		simprr 772	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ≠ 𝑦)
18	17	necomd 2988	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ≠ 𝐴)
19		ishlg.k	. . . . 5 ⊢ 𝐾 = (hlG‘𝐺)
20		hltr.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
21	20	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷 ∈ 𝑃)
22		hlcgreulem.1	. . . . . . 7 ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝐷)
23	1, 3, 19, 13, 20, 6, 4, 22	hlcomd 28588	. . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑋)
24	23	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑋)
25		simprl 770	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝐷𝐼𝑦))
26	1, 3, 19, 21, 14, 12, 5, 7, 24, 25	btwnhl 28598	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑋𝐼𝑦))
27	1, 2, 3, 5, 14, 7, 12, 26	tgbtwncom 28472	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑦𝐼𝑋))
28		hlcgreulem.2	. . . . . . 7 ⊢ (𝜑 → 𝑌(𝐾‘𝐴)𝐷)
29	1, 3, 19, 15, 20, 6, 4, 28	hlcomd 28588	. . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑌)
30	29	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑌)
31	1, 3, 19, 21, 16, 12, 5, 7, 30, 25	btwnhl 28598	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑌𝐼𝑦))
32	1, 2, 3, 5, 16, 7, 12, 31	tgbtwncom 28472	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑦𝐼𝑌))
33		hlcgreulem.3	. . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶))
34	33	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (𝐴 − 𝑋) = (𝐵 − 𝐶))
35		hlcgreulem.4	. . . 4 ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶))
36	35	ad2antrr 726	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (𝐴 − 𝑌) = (𝐵 − 𝐶))
37	1, 2, 3, 5, 7, 9, 11, 12, 14, 16, 18, 27, 32, 34, 36	tgsegconeq 28470	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 = 𝑌)
38	1	fvexi 6895	. . . . 5 ⊢ 𝑃 ∈ V
39	38	a1i 11	. . . 4 ⊢ (𝜑 → 𝑃 ∈ V)
40		hlcgrex.2	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
41	39, 8, 10, 40	nehash2 14497	. . 3 ⊢ (𝜑 → 2 ≤ (♯‘𝑃))
42	1, 2, 3, 4, 20, 6, 41	tgbtwndiff 28490	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦))
43	37, 42	r19.29a 3149	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 Vcvv 3464 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 Basecbs 17233 distcds 17285 TarskiGcstrkg 28411 Itvcitv 28417 hlGchlg 28584

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-oadd 8489 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9920 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-n0 12507 df-xnn0 12580 df-z 12594 df-uz 12858 df-fz 13530 df-hash 14354 df-trkgc 28432 df-trkgb 28433 df-trkgcb 28434 df-trkg 28437 df-hlg 28585

This theorem is referenced by: hlcgreu 28602 iscgra1 28794

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