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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 767 | . . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝜑) | |
| 2 | simpr 484 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) | |
| 3 | ttgbtwnid.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ ℂMod) | |
| 4 | clmlmod 25100 | . . . . . . . . 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 2737 | . . . . . . . . 9 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
| 9 | ttgitvval.m | . . . . . . . . 9 ⊢ − = (-g‘𝐻) | |
| 10 | 7, 8, 9 | lmodsubid 20920 | . . . . . . . 8 ⊢ ((𝐻 ∈ LMod ∧ 𝑋 ∈ 𝑃) → (𝑋 − 𝑋) = (0g‘𝐻)) | 
| 11 | 5, 6, 10 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (𝑋 − 𝑋) = (0g‘𝐻)) | 
| 12 | 11 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → (𝑋 − 𝑋) = (0g‘𝐻)) | 
| 13 | 12 | oveq2d 7447 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → (𝑘 · (𝑋 − 𝑋)) = (𝑘 · (0g‘𝐻))) | 
| 14 | 5 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝐻 ∈ LMod) | 
| 15 | ttgbtwnid.2 | . . . . . . . 8 ⊢ (𝜑 → (0[,]1) ⊆ 𝑅) | |
| 16 | 15 | ad2antrr 726 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → (0[,]1) ⊆ 𝑅) | 
| 17 | simplr 769 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝑘 ∈ (0[,]1)) | |
| 18 | 16, 17 | sseldd 3984 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝑘 ∈ 𝑅) | 
| 19 | eqid 2737 | . . . . . . 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 20894 | . . . . . 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 2781 | . . . 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 20919 | . . . . . 6 ⊢ ((𝐻 ∈ LMod ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑃) → ((𝑌 − 𝑋) = (0g‘𝐻) ↔ 𝑌 = 𝑋)) | 
| 27 | 5, 25, 6, 26 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → ((𝑌 − 𝑋) = (0g‘𝐻) ↔ 𝑌 = 𝑋)) | 
| 28 | 27 | biimpa 476 | . . . 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 2743 | . 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 28897 | . . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑋) ↔ ∃𝑘 ∈ (0[,]1)(𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋)))) | 
| 35 | 31, 34 | mpbid 232 | . 2 ⊢ (𝜑 → ∃𝑘 ∈ (0[,]1)(𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) | 
| 36 | 30, 35 | r19.29a 3162 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 [,]cicc 13390 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 -gcsg 18953 LModclmod 20858 ℂModcclm 25095 Itvcitv 28441 toTGcttg 28881 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-lmod 20860 df-clm 25096 df-itv 28443 df-lng 28444 df-ttg 28882 | 
| This theorem is referenced by: (None) | 
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