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Theorem lnxfr 29030

Description: Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)

**Hypotheses**
Ref	Expression
tglngval.p	⊢ 𝑃 = (Base‘𝐺)
tglngval.l	⊢ 𝐿 = (LineG‘𝐺)
tglngval.i	⊢ 𝐼 = (Itv‘𝐺)
tglngval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglngval.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tglngval.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgcolg.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
lnxfr.r	⊢ ∼ = (cgrG‘𝐺)
lnxfr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
lnxfr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
lnxfr.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
lnxfr.1	⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
lnxfr.2	⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)

**Assertion**
Ref	Expression
lnxfr	⊢ (𝜑 → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))

**Proof of Theorem lnxfr**
Step	Hyp	Ref	Expression
1		tglngval.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		tglngval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
3		tglngval.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		tglngval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐺 ∈ TarskiG)
6		lnxfr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐴 ∈ 𝑃)
8		lnxfr.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
9	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐶 ∈ 𝑃)
10		lnxfr.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐵 ∈ 𝑃)
12		eqid 2760	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
13		lnxfr.r	. . . 4 ⊢ ∼ = (cgrG‘𝐺)
14		tglngval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
15	14	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋 ∈ 𝑃)
16		tglngval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
17	16	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ 𝑃)
18		tgcolg.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
19	18	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑍 ∈ 𝑃)
20		lnxfr.2	. . . . 5 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)
21	20	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)
22		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ (𝑋𝐼𝑍))
23	1, 12, 3, 13, 5, 15, 17, 19, 7, 11, 9, 21, 22	tgbtwnxfr 28994	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐵 ∈ (𝐴𝐼𝐶))
24	1, 2, 3, 5, 7, 9, 11, 23	btwncolg1 29019	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
25	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐺 ∈ TarskiG)
26	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐴 ∈ 𝑃)
27	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐶 ∈ 𝑃)
28	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐵 ∈ 𝑃)
29	16	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑌 ∈ 𝑃)
30	14	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑋 ∈ 𝑃)
31	18	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑍 ∈ 𝑃)
32	20	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)
33	1, 12, 3, 13, 25, 30, 29, 31, 26, 28, 27, 32	cgr3swap12 28987	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → ⟨“𝑌𝑋𝑍”⟩ ∼ ⟨“𝐵𝐴𝐶”⟩)
34		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑋 ∈ (𝑌𝐼𝑍))
35	1, 12, 3, 13, 25, 29, 30, 31, 28, 26, 27, 33, 34	tgbtwnxfr 28994	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐴 ∈ (𝐵𝐼𝐶))
36	1, 2, 3, 25, 26, 27, 28, 35	btwncolg2 29020	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
37	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐺 ∈ TarskiG)
38	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐴 ∈ 𝑃)
39	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ 𝑃)
40	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐵 ∈ 𝑃)
41	14	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝑃)
42	18	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ 𝑃)
43	16	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝑃)
44	20	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)
45	1, 12, 3, 13, 37, 41, 43, 42, 38, 40, 39, 44	cgr3swap23 28988	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝐶𝐵”⟩)
46		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐼𝑌))
47	1, 12, 3, 13, 37, 41, 42, 43, 38, 39, 40, 45, 46	tgbtwnxfr 28994	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ (𝐴𝐼𝐵))
48	1, 2, 3, 37, 38, 39, 40, 47	btwncolg3 29021	. 2 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
49		lnxfr.1	. . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍))
50	1, 2, 3, 4, 14, 18, 16	tgcolg 29018	. . 3 ⊢ (𝜑 → ((𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍) ↔ (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))))
51	49, 50	mpbid 235	. 2 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))
52	24, 36, 48, 51	mpjao3dan 1459	1 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 = wceq 1569 ∈ wcel 2144 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 ⟨“cs3 14994 Basecbs 17388 distcds 17438 TarskiGcstrkg 28890 Itvcitv 28896 LineGclng 28897 cgrGccgrg 28974

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-oadd 8481 df-er 8718 df-pm 8851 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-dju 9983 df-card 10021 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-n0 12608 df-xnn0 12681 df-z 12695 df-uz 12967 df-fz 13641 df-fzo 13790 df-hash 14476 df-word 14660 df-concat 14717 df-s1 14744 df-s2 15000 df-s3 15001 df-trkgc 28911 df-trkgb 28912 df-trkgcb 28913 df-trkg 28916 df-cgrg 28975

This theorem is used by: symquadlem 29162 midexlem 29165 trgcopy 29312

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