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Mirrors > Home > MPE Home > Th. List > ttgbtwnid | Structured version Visualization version GIF version |
Description: Any subcomplex module equipped with the betweenness operation fulfills the identity of betweenness (Axiom A6). (Contributed by Thierry Arnoux, 26-Mar-2019.) |
Ref | Expression |
---|---|
ttgval.n | ⊢ 𝐺 = (toTG‘𝐻) |
ttgitvval.i | ⊢ 𝐼 = (Itv‘𝐺) |
ttgitvval.b | ⊢ 𝑃 = (Base‘𝐻) |
ttgitvval.m | ⊢ − = (-g‘𝐻) |
ttgitvval.s | ⊢ · = ( ·𝑠 ‘𝐻) |
ttgelitv.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
ttgelitv.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
ttgbtwnid.r | ⊢ 𝑅 = (Base‘(Scalar‘𝐻)) |
ttgbtwnid.2 | ⊢ (𝜑 → (0[,]1) ⊆ 𝑅) |
ttgbtwnid.1 | ⊢ (𝜑 → 𝐻 ∈ ℂMod) |
ttgbtwnid.y | ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑋)) |
Ref | Expression |
---|---|
ttgbtwnid | ⊢ (𝜑 → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 763 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝜑) | |
2 | simpr 485 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) | |
3 | ttgbtwnid.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ ℂMod) | |
4 | clmlmod 23598 | . . . . . . . . 9 ⊢ (𝐻 ∈ ℂMod → 𝐻 ∈ LMod) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ LMod) |
6 | ttgelitv.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
7 | ttgitvval.b | . . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐻) | |
8 | eqid 2818 | . . . . . . . . 9 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
9 | ttgitvval.m | . . . . . . . . 9 ⊢ − = (-g‘𝐻) | |
10 | 7, 8, 9 | lmodsubid 19623 | . . . . . . . 8 ⊢ ((𝐻 ∈ LMod ∧ 𝑋 ∈ 𝑃) → (𝑋 − 𝑋) = (0g‘𝐻)) |
11 | 5, 6, 10 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐻)) |
12 | 11 | ad2antrr 722 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → (𝑋 − 𝑋) = (0g‘𝐻)) |
13 | 12 | oveq2d 7161 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → (𝑘 · (𝑋 − 𝑋)) = (𝑘 · (0g‘𝐻))) |
14 | 5 | ad2antrr 722 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝐻 ∈ LMod) |
15 | ttgbtwnid.2 | . . . . . . . 8 ⊢ (𝜑 → (0[,]1) ⊆ 𝑅) | |
16 | 15 | ad2antrr 722 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → (0[,]1) ⊆ 𝑅) |
17 | simplr 765 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝑘 ∈ (0[,]1)) | |
18 | 16, 17 | sseldd 3965 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝑘 ∈ 𝑅) |
19 | eqid 2818 | . . . . . . 7 ⊢ (Scalar‘𝐻) = (Scalar‘𝐻) | |
20 | ttgitvval.s | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝐻) | |
21 | ttgbtwnid.r | . . . . . . 7 ⊢ 𝑅 = (Base‘(Scalar‘𝐻)) | |
22 | 19, 20, 21, 8 | lmodvs0 19597 | . . . . . 6 ⊢ ((𝐻 ∈ LMod ∧ 𝑘 ∈ 𝑅) → (𝑘 · (0g‘𝐻)) = (0g‘𝐻)) |
23 | 14, 18, 22 | syl2anc 584 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → (𝑘 · (0g‘𝐻)) = (0g‘𝐻)) |
24 | 2, 13, 23 | 3eqtrd 2857 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → (𝑌 − 𝑋) = (0g‘𝐻)) |
25 | ttgelitv.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
26 | 7, 8, 9 | lmodsubeq0 19622 | . . . . . 6 ⊢ ((𝐻 ∈ LMod ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑃) → ((𝑌 − 𝑋) = (0g‘𝐻) ↔ 𝑌 = 𝑋)) |
27 | 5, 25, 6, 26 | syl3anc 1363 | . . . . 5 ⊢ (𝜑 → ((𝑌 − 𝑋) = (0g‘𝐻) ↔ 𝑌 = 𝑋)) |
28 | 27 | biimpa 477 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 − 𝑋) = (0g‘𝐻)) → 𝑌 = 𝑋) |
29 | 1, 24, 28 | syl2anc 584 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝑌 = 𝑋) |
30 | 29 | eqcomd 2824 | . 2 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝑋 = 𝑌) |
31 | ttgbtwnid.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑋)) | |
32 | ttgval.n | . . . 4 ⊢ 𝐺 = (toTG‘𝐻) | |
33 | ttgitvval.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
34 | 32, 33, 7, 9, 20, 6, 6, 3, 25 | ttgelitv 26596 | . . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑋) ↔ ∃𝑘 ∈ (0[,]1)(𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋)))) |
35 | 31, 34 | mpbid 233 | . 2 ⊢ (𝜑 → ∃𝑘 ∈ (0[,]1)(𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) |
36 | 30, 35 | r19.29a 3286 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 [,]cicc 12729 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 0gc0g 16701 -gcsg 18043 LModclmod 19563 ℂModcclm 23593 Itvcitv 26149 toTGcttg 26586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-dec 12087 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mgp 19169 df-ring 19228 df-lmod 19565 df-clm 23594 df-itv 26151 df-lng 26152 df-ttg 26587 |
This theorem is referenced by: (None) |
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