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Theorem lnxfr 28908

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 28872	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐵 ∈ (𝐴𝐼𝐶))
24	1, 2, 3, 5, 7, 9, 11, 23	btwncolg1 28897	. 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 28865	. . . 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 28872	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐴 ∈ (𝐵𝐼𝐶))
36	1, 2, 3, 25, 26, 27, 28, 35	btwncolg2 28898	. 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 28866	. . . 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 28872	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ (𝐴𝐼𝐵))
48	1, 2, 3, 37, 38, 39, 40, 47	btwncolg3 28899	. 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 28896	. . 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 1570 ∈ wcel 2145 class class class wbr 5103 ‘cfv 6533 (class class class)co 7413 ⟨“cs3 14913 Basecbs 17301 distcds 17351 TarskiGcstrkg 28768 Itvcitv 28774 LineGclng 28775 cgrGccgrg 28852

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 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

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 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 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-tp 4589 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 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-oadd 8459 df-er 8696 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-dju 9906 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12888 df-fz 13562 df-fzo 13710 df-hash 14395 df-word 14579 df-concat 14636 df-s1 14663 df-s2 14919 df-s3 14920 df-trkgc 28789 df-trkgb 28790 df-trkgcb 28791 df-trkg 28794 df-cgrg 28853

This theorem is used by: symquadlem 29040 midexlem 29043 trgcopy 29190

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