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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 25114 | . . . . . . . . 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 2735 | . . . . . . . . 9 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
9 | ttgitvval.m | . . . . . . . . 9 ⊢ − = (-g‘𝐻) | |
10 | 7, 8, 9 | lmodsubid 20937 | . . . . . . . 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 3996 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0[,]1)) ∧ (𝑌 − 𝑋) = (𝑘 · (𝑋 − 𝑋))) → 𝑘 ∈ 𝑅) |
19 | eqid 2735 | . . . . . . 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 20911 | . . . . . 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 2779 | . . . 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 20936 | . . . . . 6 ⊢ ((𝐻 ∈ LMod ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑃) → ((𝑌 − 𝑋) = (0g‘𝐻) ↔ 𝑌 = 𝑋)) |
27 | 5, 25, 6, 26 | syl3anc 1370 | . . . . 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 2741 | . 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 28912 | . . 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 3160 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 [,]cicc 13387 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 -gcsg 18966 LModclmod 20875 ℂModcclm 25109 Itvcitv 28456 toTGcttg 28896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-dec 12732 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-lmod 20877 df-clm 25110 df-itv 28458 df-lng 28459 df-ttg 28897 |
This theorem is referenced by: (None) |
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