![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 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 |
Copyright terms: Public domain | W3C validator |