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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dih0bN | Structured version Visualization version GIF version |
Description: A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dih0b.b | ⊢ 𝐵 = (Base‘𝐾) |
dih0b.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dih0b.o | ⊢ 0 = (0.‘𝐾) |
dih0b.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dih0b.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dih0b.z | ⊢ 𝑍 = (0g‘𝑈) |
dih0b.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dih0b.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
dih0bN | ⊢ (𝜑 → (𝑋 = 0 ↔ (𝐼‘𝑋) = {𝑍})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dih0b.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dih0b.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | 1 | simpld 482 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ HL) |
4 | hlop 35169 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
5 | dih0b.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
6 | dih0b.o | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
7 | 5, 6 | op0cl 34991 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
8 | 3, 4, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) |
9 | dih0b.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
10 | dih0b.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
11 | 5, 9, 10 | dih11 37073 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → ((𝐼‘𝑋) = (𝐼‘ 0 ) ↔ 𝑋 = 0 )) |
12 | 1, 2, 8, 11 | syl3anc 1476 | . 2 ⊢ (𝜑 → ((𝐼‘𝑋) = (𝐼‘ 0 ) ↔ 𝑋 = 0 )) |
13 | dih0b.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
14 | dih0b.z | . . . . 5 ⊢ 𝑍 = (0g‘𝑈) | |
15 | 6, 9, 10, 13, 14 | dih0 37088 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {𝑍}) |
16 | 1, 15 | syl 17 | . . 3 ⊢ (𝜑 → (𝐼‘ 0 ) = {𝑍}) |
17 | 16 | eqeq2d 2781 | . 2 ⊢ (𝜑 → ((𝐼‘𝑋) = (𝐼‘ 0 ) ↔ (𝐼‘𝑋) = {𝑍})) |
18 | 12, 17 | bitr3d 270 | 1 ⊢ (𝜑 → (𝑋 = 0 ↔ (𝐼‘𝑋) = {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 {csn 4317 ‘cfv 6030 Basecbs 16064 0gc0g 16308 0.cp0 17245 OPcops 34979 HLchlt 35157 LHypclh 35791 DVecHcdvh 36886 DIsoHcdih 37036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-riotaBAD 34759 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-tpos 7508 df-undef 7555 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7900 df-map 8015 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-5 11288 df-6 11289 df-n0 11500 df-z 11585 df-uz 11894 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-sca 16165 df-vsca 16166 df-0g 16310 df-preset 17136 df-poset 17154 df-plt 17166 df-lub 17182 df-glb 17183 df-join 17184 df-meet 17185 df-p0 17247 df-p1 17248 df-lat 17254 df-clat 17316 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-grp 17633 df-minusg 17634 df-sbg 17635 df-subg 17799 df-cntz 17957 df-lsm 18258 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-invr 18880 df-dvr 18891 df-drng 18959 df-lmod 19075 df-lss 19143 df-lsp 19185 df-lvec 19316 df-oposet 34983 df-ol 34985 df-oml 34986 df-covers 35073 df-ats 35074 df-atl 35105 df-cvlat 35129 df-hlat 35158 df-llines 35305 df-lplanes 35306 df-lvols 35307 df-lines 35308 df-psubsp 35310 df-pmap 35311 df-padd 35603 df-lhyp 35795 df-laut 35796 df-ldil 35911 df-ltrn 35912 df-trl 35967 df-tendo 36563 df-edring 36565 df-disoa 36837 df-dvech 36887 df-dib 36947 df-dic 36981 df-dih 37037 |
This theorem is referenced by: (None) |
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