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Theorem hlcgreulem 29065

Description: Lemma for hlcgreu 29066. (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 739	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐺 ∈ TarskiG)
6		ishlg.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad2antrr 739	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ 𝑃)
8		ishlg.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	ad2antrr 739	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐵 ∈ 𝑃)
10		ishlg.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	ad2antrr 739	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐶 ∈ 𝑃)
12		simplr 781	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑦 ∈ 𝑃)
13		hlcgreulem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
14	13	ad2antrr 739	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 ∈ 𝑃)
15		hlcgreulem.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
16	15	ad2antrr 739	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑌 ∈ 𝑃)
17		simprr 785	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ≠ 𝑦)
18	17	necomd 3011	. . 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 739	. . . . 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 29052	. . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑋)
24	23	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑋)
25		simprl 783	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝐷𝐼𝑦))
26	1, 3, 19, 21, 14, 12, 5, 7, 24, 25	btwnhl 29062	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑋𝐼𝑦))
27	1, 2, 3, 5, 14, 7, 12, 26	tgbtwncom 28933	. . 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 29052	. . . . . 6 ⊢ (𝜑 → 𝐷(𝐾‘𝐴)𝑌)
30	29	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐷(𝐾‘𝐴)𝑌)
31	1, 3, 19, 21, 16, 12, 5, 7, 30, 25	btwnhl 29062	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑌𝐼𝑦))
32	1, 2, 3, 5, 16, 7, 12, 31	tgbtwncom 28933	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝐴 ∈ (𝑦𝐼𝑌))
33		hlcgreulem.3	. . . 4 ⊢ (𝜑 → (𝐴 − 𝑋) = (𝐵 − 𝐶))
34	33	ad2antrr 739	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → (𝐴 − 𝑋) = (𝐵 − 𝐶))
35		hlcgreulem.4	. . . 4 ⊢ (𝜑 → (𝐴 − 𝑌) = (𝐵 − 𝐶))
36	35	ad2antrr 739	. . 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 28930	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦)) → 𝑋 = 𝑌)
38	1	fvexi 6891	. . . . 5 ⊢ 𝑃 ∈ V
39	38	a1i 11	. . . 4 ⊢ (𝜑 → 𝑃 ∈ V)
40		hlcgrex.2	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
41	39, 8, 10, 40	nehash2 14599	. . 3 ⊢ (𝜑 → 2 ≤ (♯‘𝑃))
42	1, 2, 3, 4, 20, 6, 41	tgbtwndiff 28951	. 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝐴 ∈ (𝐷𝐼𝑦) ∧ 𝐴 ≠ 𝑦))
43	37, 42	r19.29a 3171	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 Vcvv 3451 class class class wbr 5103 ‘cfv 6531 (class class class)co 7412 Basecbs 17367 distcds 17417 TarskiGcstrkg 28871 Itvcitv 28877 hlGchlg 29045

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-oadd 8464 df-er 8701 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9963 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-n0 12588 df-xnn0 12661 df-z 12675 df-uz 12947 df-fz 13621 df-hash 14455 df-trkgc 28892 df-trkgb 28893 df-trkgcb 28894 df-trkg 28897 df-hlg 29046

This theorem is used by: hlcgreu 29066 hlcgreq 29067 iscgra1 29299

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