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Theorem tgbtwnxfr 28927

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 739	. . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐺 ∈ TarskiG)
6		simplr 781	. . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒 ∈ 𝑃)
7		tgbtwnxfr.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
8	7	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐸 ∈ 𝑃)
9		tgbtwnxfr.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
10	9	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐷 ∈ 𝑃)
11		tgbtwnxfr.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
12	11	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐹 ∈ 𝑃)
13		simprl 783	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒 ∈ (𝐷𝐼𝐹))
14		eqidd 2761	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝐷 − 𝐹) = (𝐷 − 𝐹))
15		eqidd 2761	. . . . 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 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐴 ∈ 𝑃)
19		tgbtwnxfr.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
20	19	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐵 ∈ 𝑃)
21		tgbtwnxfr.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
22	21	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝐶 ∈ 𝑃)
23		simprr 785	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)
24	1, 2, 3, 16, 5, 18, 20, 22, 10, 6, 12, 23	trgcgrcom 28925	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝑒𝐹”⟩ ∼ ⟨“𝐴𝐵𝐶”⟩)
25		tgbtwnxfr.2	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
26	25	ad2antrr 739	. . . . . . . 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 28926	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝑒𝐹”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
28	1, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27	trgcgrcom 28925	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝐸𝐹”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)
29	1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28	cgr3simp1 28917	. . . . 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 28918	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝐸 − 𝐹) = (𝑒 − 𝐹))
31	1, 2, 3, 5, 8, 12, 6, 12, 30	tgcgrcomlr 28876	. . . . 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 28905	. . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝑒 − 𝐸) = (𝑒 − 𝑒))
33	1, 2, 3, 5, 6, 8, 6, 32	axtgcgrid 28859	. . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → 𝑒 = 𝐸)
34	33, 13	eqeltrrd 2861	. 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 28919	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷))
37	1, 2, 3, 4, 21, 17, 11, 9, 36	tgcgrcomlr 28876	. . 3 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))
38	1, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37	tgcgrxfr 28915	. 2 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩))
39	34, 38	r19.29a 3170	1 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 class class class wbr 5102 ‘cfv 6527 (class class class)co 7408 ⟨“cs3 14961 Basecbs 17349 distcds 17399 TarskiGcstrkg 28823 Itvcitv 28829 cgrGccgrg 28907

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 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249

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 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 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 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-oadd 8458 df-er 8695 df-pm 8828 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-dju 9954 df-card 9992 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-nn 12306 df-2 12375 df-3 12376 df-n0 12577 df-xnn0 12650 df-z 12664 df-uz 12936 df-fz 13610 df-fzo 13758 df-hash 14443 df-word 14627 df-concat 14684 df-s1 14711 df-s2 14967 df-s3 14968 df-trkgc 28844 df-trkgb 28845 df-trkgcb 28846 df-trkg 28849 df-cgrg 28908

This theorem is used by: lnxfr 28963 tgfscgr 28965 legov 28982 legov2 28983 legtrd 28986 mirbtwni 29077 cgrabtwn 29268 cgrahl 29269

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