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Mirrors > Home > MPE Home > Th. List > Mathboxes > dih11 | Structured version Visualization version GIF version |
Description: The isomorphism H is one-to-one. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.) |
Ref | Expression |
---|---|
dih11.b | ⊢ 𝐵 = (Base‘𝐾) |
dih11.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dih11.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dih11 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘𝑋) = (𝐼‘𝑌) ↔ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqss 3921 | . 2 ⊢ ((𝐼‘𝑋) = (𝐼‘𝑌) ↔ ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ (𝐼‘𝑌) ⊆ (𝐼‘𝑋))) | |
2 | dih11.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
3 | eqid 2737 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | dih11.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dih11.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | 2, 3, 4, 5 | dihord 39020 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋(le‘𝐾)𝑌)) |
7 | 2, 3, 4, 5 | dihord 39020 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐼‘𝑌) ⊆ (𝐼‘𝑋) ↔ 𝑌(le‘𝐾)𝑋)) |
8 | 7 | 3com23 1128 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘𝑌) ⊆ (𝐼‘𝑋) ↔ 𝑌(le‘𝐾)𝑋)) |
9 | 6, 8 | anbi12d 634 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ (𝐼‘𝑌) ⊆ (𝐼‘𝑋)) ↔ (𝑋(le‘𝐾)𝑌 ∧ 𝑌(le‘𝐾)𝑋))) |
10 | simp1l 1199 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ HL) | |
11 | 10 | hllatd 37120 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) |
12 | 2, 3 | latasymb 17953 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(le‘𝐾)𝑌 ∧ 𝑌(le‘𝐾)𝑋) ↔ 𝑋 = 𝑌)) |
13 | 11, 12 | syld3an1 1412 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋(le‘𝐾)𝑌 ∧ 𝑌(le‘𝐾)𝑋) ↔ 𝑋 = 𝑌)) |
14 | 9, 13 | bitrd 282 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ∧ (𝐼‘𝑌) ⊆ (𝐼‘𝑋)) ↔ 𝑋 = 𝑌)) |
15 | 1, 14 | syl5bb 286 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘𝑋) = (𝐼‘𝑌) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ⊆ wss 3871 class class class wbr 5058 ‘cfv 6385 Basecbs 16765 lecple 16814 Latclat 17942 HLchlt 37106 LHypclh 37740 DIsoHcdih 38984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-resscn 10791 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-mulcom 10798 ax-addass 10799 ax-mulass 10800 ax-distr 10801 ax-i2m1 10802 ax-1ne0 10803 ax-1rid 10804 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 ax-pre-lttri 10808 ax-pre-lttrn 10809 ax-pre-ltadd 10810 ax-pre-mulgt0 10811 ax-riotaBAD 36709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-int 4865 df-iun 4911 df-iin 4912 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-riota 7175 df-ov 7221 df-oprab 7222 df-mpo 7223 df-om 7650 df-1st 7766 df-2nd 7767 df-tpos 7973 df-undef 8020 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-1o 8207 df-er 8396 df-map 8515 df-en 8632 df-dom 8633 df-sdom 8634 df-fin 8635 df-pnf 10874 df-mnf 10875 df-xr 10876 df-ltxr 10877 df-le 10878 df-sub 11069 df-neg 11070 df-nn 11836 df-2 11898 df-3 11899 df-4 11900 df-5 11901 df-6 11902 df-n0 12096 df-z 12182 df-uz 12444 df-fz 13101 df-struct 16705 df-sets 16722 df-slot 16740 df-ndx 16750 df-base 16766 df-ress 16790 df-plusg 16820 df-mulr 16821 df-sca 16823 df-vsca 16824 df-0g 16951 df-proset 17807 df-poset 17825 df-plt 17841 df-lub 17857 df-glb 17858 df-join 17859 df-meet 17860 df-p0 17936 df-p1 17937 df-lat 17943 df-clat 18010 df-mgm 18119 df-sgrp 18168 df-mnd 18179 df-submnd 18224 df-grp 18373 df-minusg 18374 df-sbg 18375 df-subg 18545 df-cntz 18716 df-lsm 19030 df-cmn 19177 df-abl 19178 df-mgp 19510 df-ur 19522 df-ring 19569 df-oppr 19646 df-dvdsr 19664 df-unit 19665 df-invr 19695 df-dvr 19706 df-drng 19774 df-lmod 19906 df-lss 19974 df-lsp 20014 df-lvec 20145 df-oposet 36932 df-ol 36934 df-oml 36935 df-covers 37022 df-ats 37023 df-atl 37054 df-cvlat 37078 df-hlat 37107 df-llines 37254 df-lplanes 37255 df-lvols 37256 df-lines 37257 df-psubsp 37259 df-pmap 37260 df-padd 37552 df-lhyp 37744 df-laut 37745 df-ldil 37860 df-ltrn 37861 df-trl 37915 df-tendo 38511 df-edring 38513 df-disoa 38785 df-dvech 38835 df-dib 38895 df-dic 38929 df-dih 38985 |
This theorem is referenced by: dihf11 39023 dihcnv11 39031 dih0bN 39037 dihlspsnat 39089 dihatexv 39094 dihatexv2 39095 dihmeet2 39102 dochvalr3 39119 djhljjN 39158 dihjat5N 39193 |
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