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 758 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐺 ∈ TarskiG) |
6 | simplr 788 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 ∈ 𝑃) | |
7 | tgbtwnxfr.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
8 | 7 | ad2antrr 758 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐸 ∈ 𝑃) |
9 | tgbtwnxfr.d | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
10 | 9 | ad2antrr 758 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐷 ∈ 𝑃) |
11 | tgbtwnxfr.f | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
12 | 11 | ad2antrr 758 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐹 ∈ 𝑃) |
13 | simprl 790 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 ∈ (𝐷𝐼𝐹)) | |
14 | eqidd 2611 | . . . . 5 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝐷 − 𝐹) = (𝐷 − 𝐹)) | |
15 | eqidd 2611 | . . . . 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 758 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐴 ∈ 𝑃) |
19 | tgbtwnxfr.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
20 | 19 | ad2antrr 758 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐵 ∈ 𝑃) |
21 | tgbtwnxfr.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
22 | 21 | ad2antrr 758 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝐶 ∈ 𝑃) |
23 | simprr 792 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉) | |
24 | 1, 2, 3, 16, 5, 18, 20, 22, 10, 6, 12, 23 | trgcgrcom 25223 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝑒𝐹”〉 ∼ 〈“𝐴𝐵𝐶”〉) |
25 | tgbtwnxfr.2 | . . . . . . . . 9 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝐸𝐹”〉) | |
26 | 25 | ad2antrr 758 | . . . . . . . 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 25224 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝑒𝐹”〉 ∼ 〈“𝐷𝐸𝐹”〉) |
28 | 1, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27 | trgcgrcom 25223 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 〈“𝐷𝐸𝐹”〉 ∼ 〈“𝐷𝑒𝐹”〉) |
29 | 1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28 | cgr3simp1 25215 | . . . . 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 25216 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝐸 − 𝐹) = (𝑒 − 𝐹)) |
31 | 1, 2, 3, 5, 8, 12, 6, 12, 30 | tgcgrcomlr 25175 | . . . . 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 25203 | . . . 4 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → (𝑒 − 𝐸) = (𝑒 − 𝑒)) |
33 | 1, 2, 3, 5, 6, 8, 6, 32 | axtgcgrid 25162 | . . 3 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) → 𝑒 = 𝐸) |
34 | 33, 13 | eqeltrrd 2689 | . 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 25217 | . . . 4 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐹 − 𝐷)) |
37 | 1, 2, 3, 4, 21, 17, 11, 9, 36 | tgcgrcomlr 25175 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐷 − 𝐹)) |
38 | 1, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37 | tgcgrxfr 25213 | . 2 ⊢ (𝜑 → ∃𝑒 ∈ 𝑃 (𝑒 ∈ (𝐷𝐼𝐹) ∧ 〈“𝐴𝐵𝐶”〉 ∼ 〈“𝐷𝑒𝐹”〉)) |
39 | 34, 38 | r19.29a 3060 | 1 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 〈“cs3 13438 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 cgrGccgrg 25205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 df-cgrg 25206 |
This theorem is referenced by: lnxfr 25261 tgfscgr 25263 legov 25280 legov2 25281 legtrd 25284 mirbtwni 25366 cgrabtwn 25517 cgrahl 25518 |
Copyright terms: Public domain | W3C validator |