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Theorem lnxfr 28500

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 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐺 ∈ TarskiG)
6		lnxfr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐴 ∈ 𝑃)
8		lnxfr.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
9	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐶 ∈ 𝑃)
10		lnxfr.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐵 ∈ 𝑃)
12		eqid 2730	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
13		lnxfr.r	. . . 4 ⊢ ∼ = (cgrG‘𝐺)
14		tglngval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
15	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋 ∈ 𝑃)
16		tglngval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
17	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ 𝑃)
18		tgcolg.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
19	18	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑍 ∈ 𝑃)
20		lnxfr.2	. . . . 5 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)
21	20	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)
22		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ (𝑋𝐼𝑍))
23	1, 12, 3, 13, 5, 15, 17, 19, 7, 11, 9, 21, 22	tgbtwnxfr 28464	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐵 ∈ (𝐴𝐼𝐶))
24	1, 2, 3, 5, 7, 9, 11, 23	btwncolg1 28489	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
25	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐺 ∈ TarskiG)
26	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐴 ∈ 𝑃)
27	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐶 ∈ 𝑃)
28	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐵 ∈ 𝑃)
29	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑌 ∈ 𝑃)
30	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑋 ∈ 𝑃)
31	18	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑍 ∈ 𝑃)
32	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)
33	1, 12, 3, 13, 25, 30, 29, 31, 26, 28, 27, 32	cgr3swap12 28457	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → ⟨“𝑌𝑋𝑍”⟩ ∼ ⟨“𝐵𝐴𝐶”⟩)
34		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝑋 ∈ (𝑌𝐼𝑍))
35	1, 12, 3, 13, 25, 29, 30, 31, 28, 26, 27, 33, 34	tgbtwnxfr 28464	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → 𝐴 ∈ (𝐵𝐼𝐶))
36	1, 2, 3, 25, 26, 27, 28, 35	btwncolg2 28490	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑌𝐼𝑍)) → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))
37	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐺 ∈ TarskiG)
38	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐴 ∈ 𝑃)
39	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ 𝑃)
40	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐵 ∈ 𝑃)
41	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝑃)
42	18	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ 𝑃)
43	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝑃)
44	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)
45	1, 12, 3, 13, 37, 41, 43, 42, 38, 40, 39, 44	cgr3swap23 28458	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → ⟨“𝑋𝑍𝑌”⟩ ∼ ⟨“𝐴𝐶𝐵”⟩)
46		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐼𝑌))
47	1, 12, 3, 13, 37, 41, 42, 43, 38, 39, 40, 45, 46	tgbtwnxfr 28464	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐶 ∈ (𝐴𝐼𝐵))
48	1, 2, 3, 37, 38, 39, 40, 47	btwncolg3 28491	. 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 28488	. . 3 ⊢ (𝜑 → ((𝑌 ∈ (𝑋𝐿𝑍) ∨ 𝑋 = 𝑍) ↔ (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))))
51	49, 50	mpbid 232	. 2 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))
52	24, 36, 48, 51	mpjao3dan 1434	1 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐿𝐶) ∨ 𝐴 = 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 = wceq 1540 ∈ wcel 2109 class class class wbr 5110 ‘cfv 6514 (class class class)co 7390 ⟨“cs3 14815 Basecbs 17186 distcds 17236 TarskiGcstrkg 28361 Itvcitv 28367 LineGclng 28368 cgrGccgrg 28444

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-oadd 8441 df-er 8674 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-xnn0 12523 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-hash 14303 df-word 14486 df-concat 14543 df-s1 14568 df-s2 14821 df-s3 14822 df-trkgc 28382 df-trkgb 28383 df-trkgcb 28384 df-trkg 28387 df-cgrg 28445

This theorem is referenced by: symquadlem 28623 midexlem 28626 trgcopy 28738

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