tgbtwnxfr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tgbtwnxfr		Structured version Visualization version GIF version

Theorem tgbtwnxfr 28994

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 28992	. . . . . . . 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 28993	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝑒𝐹”⟩ ∼ ⟨“𝐷𝐸𝐹”⟩)
28	1, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27	trgcgrcom 28992	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → ⟨“𝐷𝐸𝐹”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)
29	1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28	cgr3simp1 28984	. . . . 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 28985	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝐸 − 𝐹) = (𝑒 − 𝐹))
31	1, 2, 3, 5, 8, 12, 6, 12, 30	tgcgrcomlr 28943	. . . . 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 28972	. . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ ⟨“𝐴𝐵𝐶”⟩ ∼ ⟨“𝐷𝑒𝐹”⟩)) → (𝑒 − 𝐸) = (𝑒 − 𝑒))
33	1, 2, 3, 5, 6, 8, 6, 32	axtgcgrid 28926	. . 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 28986	. . . 4 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷))
37	1, 2, 3, 4, 21, 17, 11, 9, 36	tgcgrcomlr 28943	. . 3 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹))
38	1, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37	tgcgrxfr 28982	. 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 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 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: lnxfr 29030 tgfscgr 29032 legov 29049 legov2 29050 legtrd 29053 mirbtwni 29144 cgrabtwn 29335 cgrahl 29336 cgrgptxfr 33054

W3C validator