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Theorem hlcgreulem 25512

Description: Lemma for hlcgreu 25513. (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 762	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐺 ∈ TarskiG)
6		ishlg.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 762	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ 𝑃)
8		ishlg.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad2antrr 762	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐵 ∈ 𝑃)
10		ishlg.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	ad2antrr 762	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐶 ∈ 𝑃)
12		simplr 792	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ∈ 𝑃)
13		hlcgreulem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
14	13	ad2antrr 762	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 ∈ 𝑃)
15		hlcgreulem.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
16	15	ad2antrr 762	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑌 ∈ 𝑃)
17		simprr 796	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ≠ 𝑦)
18	17	necomd 2849	. . 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 762	. . . . 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 25499	. . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑋)
24	23	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑋)
25		simprl 794	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝐷𝐼𝑦))
26	1, 3, 19, 21, 14, 12, 5, 7, 24, 25	btwnhl 25509	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑋𝐼𝑦))
27	1, 2, 3, 5, 14, 7, 12, 26	tgbtwncom 25383	. . 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 25499	. . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑌)
30	29	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑌)
31	1, 3, 19, 21, 16, 12, 5, 7, 30, 25	btwnhl 25509	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑌𝐼𝑦))
32	1, 2, 3, 5, 16, 7, 12, 31	tgbtwncom 25383	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑦𝐼𝑌))
33		hlcgreulem.3	. . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶))
34	33	ad2antrr 762	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (𝐴 − 𝑋) = (𝐵 − 𝐶))
35		hlcgreulem.4	. . . 4 ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶))
36	35	ad2antrr 762	. . 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 25381	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 = 𝑌)
38		fvex 6201	. . . . . 6 ⊢ (Base‘𝐺) ∈ V
39	1, 38	eqeltri 2697	. . . . 5 ⊢ 𝑃 ∈ V
40	39	a1i 11	. . . 4 ⊢ (𝜑 → 𝑃 ∈ V)
41		hlcgrex.2	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
42	40, 8, 10, 41	nehash2 13256	. . 3 ⊢ (𝜑 → 2 ≤ (#‘𝑃))
43	1, 2, 3, 4, 20, 6, 42	tgbtwndiff 25401	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦))
44	37, 43	r19.29a 3078	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 hlGchlg 25495

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-hlg 25496

This theorem is referenced by: hlcgreu 25513 iscgra1 25702

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