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Mirrors > Home > MPE Home > Th. List > tgbtwnxfr | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
tgbtwnxfr | ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) |
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 717 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐺 ∈ TarskiG) |
6 | simplr 785 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 ∈ 𝑃) | |
7 | tgbtwnxfr.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
8 | 7 | ad2antrr 717 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐸 ∈ 𝑃) |
9 | tgbtwnxfr.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
10 | 9 | ad2antrr 717 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐷 ∈ 𝑃) |
11 | tgbtwnxfr.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
12 | 11 | ad2antrr 717 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐹 ∈ 𝑃) |
13 | simprl 787 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 ∈ (𝐷𝐼𝐹)) | |
14 | eqidd 2826 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝐷 − 𝐹) = (𝐷 − 𝐹)) | |
15 | eqidd 2826 | . . . . 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 717 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐴 ∈ 𝑃) |
19 | tgbtwnxfr.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
20 | 19 | ad2antrr 717 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐵 ∈ 𝑃) |
21 | tgbtwnxfr.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
22 | 21 | ad2antrr 717 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐶 ∈ 𝑃) |
23 | simprr 789 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉) | |
24 | 1, 2, 3, 16, 5, 18, 20, 22, 10, 6, 12, 23 | trgcgrcom 25840 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝑒𝐹”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
25 | tgbtwnxfr.2 | . . . . . . . . 9 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) | |
26 | 25 | ad2antrr 717 | . . . . . . . 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 25841 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝑒𝐹”〉 ∼ 〈“𝐷𝐸𝐹”〉) |
28 | 1, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27 | trgcgrcom 25840 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝐸𝐹”〉 ∼ 〈“𝐷𝑒𝐹”〉) |
29 | 1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28 | cgr3simp1 25832 | . . . . 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 25833 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝐸 − 𝐹) = (𝑒 − 𝐹)) |
31 | 1, 2, 3, 5, 8, 12, 6, 12, 30 | tgcgrcomlr 25792 | . . . . 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 25820 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝑒 − 𝐸) = (𝑒 − 𝑒)) |
33 | 1, 2, 3, 5, 6, 8, 6, 32 | axtgcgrid 25775 | . . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 = 𝐸) |
34 | 33, 13 | eqeltrrd 2907 | . 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 25834 | . . . 4 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
37 | 1, 2, 3, 4, 21, 17, 11, 9, 36 | tgcgrcomlr 25792 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
38 | 1, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37 | tgcgrxfr 25830 | . 2 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) |
39 | 34, 38 | r19.29a 3288 | 1 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 〈“cs3 13963 Basecbs 16222 distcds 16314 TarskiGcstrkg 25742 Itvcitv 25748 cgrGccgrg 25822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-xnn0 11691 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-hash 13411 df-word 13575 df-concat 13631 df-s1 13656 df-s2 13969 df-s3 13970 df-trkgc 25760 df-trkgb 25761 df-trkgcb 25762 df-trkg 25765 df-cgrg 25823 |
This theorem is referenced by: lnxfr 25878 tgfscgr 25880 legov 25897 legov2 25898 legtrd 25901 mirbtwni 25983 cgrabtwn 26134 cgrahl 26135 |
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