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Theorem hlcgreulem 28640

Description: Lemma for hlcgreu 28641. (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 769	. . 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 773	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ≠ 𝑦)
18	17	necomd 2994	. . 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 28627	. . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑋)
24	23	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑋)
25		simprl 771	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝐷𝐼𝑦))
26	1, 3, 19, 21, 14, 12, 5, 7, 24, 25	btwnhl 28637	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑋𝐼𝑦))
27	1, 2, 3, 5, 14, 7, 12, 26	tgbtwncom 28511	. . 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 28627	. . . . . 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 28637	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑌𝐼𝑦))
32	1, 2, 3, 5, 16, 7, 12, 31	tgbtwncom 28511	. . 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 28509	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 = 𝑌)
38	1	fvexi 6921	. . . . 5 ⊢ 𝑃 ∈ V
39	38	a1i 11	. . . 4 ⊢ (𝜑 → 𝑃 ∈ V)
40		hlcgrex.2	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
41	39, 8, 10, 40	nehash2 14510	. . 3 ⊢ (𝜑 → 2 ≤ (♯‘𝑃))
42	1, 2, 3, 4, 20, 6, 41	tgbtwndiff 28529	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦))
43	37, 42	r19.29a 3160	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 distcds 17307 TarskiGcstrkg 28450 Itvcitv 28456 hlGchlg 28623

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-trkgc 28471 df-trkgb 28472 df-trkgcb 28473 df-trkg 28476 df-hlg 28624

This theorem is referenced by: hlcgreu 28641 iscgra1 28833

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