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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 724 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐺 ∈ TarskiG) |
6 | simplr 767 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 ∈ 𝑃) | |
7 | tgbtwnxfr.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
8 | 7 | ad2antrr 724 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐸 ∈ 𝑃) |
9 | tgbtwnxfr.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
10 | 9 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐷 ∈ 𝑃) |
11 | tgbtwnxfr.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
12 | 11 | ad2antrr 724 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐹 ∈ 𝑃) |
13 | simprl 769 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 ∈ (𝐷𝐼𝐹)) | |
14 | eqidd 2824 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝐷 − 𝐹) = (𝐷 − 𝐹)) | |
15 | eqidd 2824 | . . . . 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 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐴 ∈ 𝑃) |
19 | tgbtwnxfr.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
20 | 19 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐵 ∈ 𝑃) |
21 | tgbtwnxfr.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
22 | 21 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐶 ∈ 𝑃) |
23 | simprr 771 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉) | |
24 | 1, 2, 3, 16, 5, 18, 20, 22, 10, 6, 12, 23 | trgcgrcom 26316 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝑒𝐹”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
25 | tgbtwnxfr.2 | . . . . . . . . 9 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) | |
26 | 25 | ad2antrr 724 | . . . . . . . 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 26317 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝑒𝐹”〉 ∼ 〈“𝐷𝐸𝐹”〉) |
28 | 1, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27 | trgcgrcom 26316 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝐸𝐹”〉 ∼ 〈“𝐷𝑒𝐹”〉) |
29 | 1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28 | cgr3simp1 26308 | . . . . 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 26309 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝐸 − 𝐹) = (𝑒 − 𝐹)) |
31 | 1, 2, 3, 5, 8, 12, 6, 12, 30 | tgcgrcomlr 26268 | . . . . 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 26296 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝑒 − 𝐸) = (𝑒 − 𝑒)) |
33 | 1, 2, 3, 5, 6, 8, 6, 32 | axtgcgrid 26251 | . . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 = 𝐸) |
34 | 33, 13 | eqeltrrd 2916 | . 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 26310 | . . . 4 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
37 | 1, 2, 3, 4, 21, 17, 11, 9, 36 | tgcgrcomlr 26268 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
38 | 1, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37 | tgcgrxfr 26306 | . 2 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) |
39 | 34, 38 | r19.29a 3291 | 1 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 〈“cs3 14206 Basecbs 16485 distcds 16576 TarskiGcstrkg 26218 Itvcitv 26224 cgrGccgrg 26298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 df-s3 14213 df-trkgc 26236 df-trkgb 26237 df-trkgcb 26238 df-trkg 26241 df-cgrg 26299 |
This theorem is referenced by: lnxfr 26354 tgfscgr 26356 legov 26373 legov2 26374 legtrd 26377 mirbtwni 26459 cgrabtwn 26614 cgrahl 26615 |
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