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Theorem tgbtwnxfr 25781

Description: A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Thierry Arnoux, 27-Apr-2019.)

**Hypotheses**
Ref	Expression
tgcgrxfr.p	⊢ 𝑃 = (Base‘𝐺)
tgcgrxfr.m	⊢ − = (dist‘𝐺)
tgcgrxfr.i	⊢ 𝐼 = (Itv‘𝐺)
tgcgrxfr.r	⊢ ∼ = (cgrG‘𝐺)
tgcgrxfr.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwnxfr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwnxfr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgbtwnxfr.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgbtwnxfr.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgbtwnxfr.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
tgbtwnxfr.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
tgbtwnxfr.2	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
tgbtwnxfr.1	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))

**Assertion**
Ref	Expression
tgbtwnxfr	⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))

Proof of Theorem tgbtwnxfr Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgcgrxfr.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		tgcgrxfr.m	. . . 4 ⊢ − = (dist‘𝐺)
3		tgcgrxfr.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		tgcgrxfr.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 718	. . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐺 ∈ TarskiG)
6		simplr 786	. . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒 ∈ 𝑃)
7		tgbtwnxfr.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
8	7	ad2antrr 718	. . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐸 ∈ 𝑃)
9		tgbtwnxfr.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
10	9	ad2antrr 718	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐷 ∈ 𝑃)
11		tgbtwnxfr.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
12	11	ad2antrr 718	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐹 ∈ 𝑃)
13		simprl 788	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒 ∈ (𝐷𝐼𝐹))
14		eqidd 2800	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝐷 − 𝐹) = (𝐷 − 𝐹))
15		eqidd 2800	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝑒 − 𝐹) = (𝑒 − 𝐹))
16		tgcgrxfr.r	. . . . . 6 ⊢ ∼ = (cgrG‘𝐺)
17		tgbtwnxfr.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
18	17	ad2antrr 718	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐴 ∈ 𝑃)
19		tgbtwnxfr.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
20	19	ad2antrr 718	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐵 ∈ 𝑃)
21		tgbtwnxfr.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
22	21	ad2antrr 718	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐶 ∈ 𝑃)
23		simprr 790	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)
24	1, 2, 3, 16, 5, 18, 20, 22, 10, 6, 12, 23	trgcgrcom 25779	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝑒𝐹”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)
25		tgbtwnxfr.2	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
26	25	ad2antrr 718	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
27	1, 2, 3, 16, 5, 10, 6, 12, 18, 20, 22, 24, 10, 8, 12, 26	cgr3tr 25780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝑒𝐹”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
28	1, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27	trgcgrcom 25779	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝐸𝐹”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)
29	1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28	cgr3simp1 25771	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝐷 − 𝐸) = (𝐷 − 𝑒))
30	1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28	cgr3simp2 25772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝐸 − 𝐹) = (𝑒 − 𝐹))
31	1, 2, 3, 5, 8, 12, 6, 12, 30	tgcgrcomlr 25731	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝐹 − 𝐸) = (𝐹 − 𝑒))
32	1, 2, 3, 5, 10, 6, 12, 8, 10, 6, 12, 6, 13, 13, 14, 15, 29, 31	tgifscgr 25759	. . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝑒 − 𝐸) = (𝑒 − 𝑒))
33	1, 2, 3, 5, 6, 8, 6, 32	axtgcgrid 25714	. . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒 = 𝐸)
34	33, 13	eqeltrrd 2879	. 2 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐸 ∈ (𝐷𝐼𝐹))
35		tgbtwnxfr.1	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
36	1, 2, 3, 16, 4, 17, 19, 21, 9, 7, 11, 25	cgr3simp3 25773	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷))
37	1, 2, 3, 4, 21, 17, 11, 9, 36	tgcgrcomlr 25731	. . 3 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))
38	1, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37	tgcgrxfr 25769	. 2 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩))
39	34, 38	r19.29a 3259	1 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 ⟨“cs3 13927 Basecbs 16184 distcds 16276 TarskiGcstrkg 25681 Itvcitv 25687 cgrGccgrg 25761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-xnn0 11653 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-hash 13371 df-word 13535 df-concat 13591 df-s1 13616 df-s2 13933 df-s3 13934 df-trkgc 25699 df-trkgb 25700 df-trkgcb 25701 df-trkg 25704 df-cgrg 25762

This theorem is referenced by: lnxfr 25817 tgfscgr 25819 legov 25836 legov2 25837 legtrd 25840 mirbtwni 25922 cgrabtwn 26073 cgrahl 26074

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