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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihprrnlem2 | Structured version Visualization version GIF version |
Description: Lemma for dihprrn 37232. (Contributed by NM, 29-Sep-2014.) |
Ref | Expression |
---|---|
dihprrn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihprrn.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihprrn.v | ⊢ 𝑉 = (Base‘𝑈) |
dihprrn.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dihprrn.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihprrn.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihprrn.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
dihprrn.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
dihprrnlem2.o | ⊢ 0 = (0g‘𝑈) |
dihprrnlem2.xz | ⊢ (𝜑 → 𝑋 ≠ 0 ) |
dihprrnlem2.yz | ⊢ (𝜑 → 𝑌 ≠ 0 ) |
Ref | Expression |
---|---|
dihprrnlem2 | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4319 | . . . 4 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
2 | 1 | fveq2i 6335 | . . 3 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
3 | dihprrn.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | eqid 2771 | . . . . 5 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | eqid 2771 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | dihprrn.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | eqid 2771 | . . . . 5 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
8 | dihprrn.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
9 | dihprrn.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | dihprrn.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | dihprrnlem2.xz | . . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 0 ) | |
12 | dihprrn.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑈) | |
13 | dihprrnlem2.o | . . . . . . 7 ⊢ 0 = (0g‘𝑈) | |
14 | dihprrn.n | . . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈) | |
15 | 5, 3, 6, 12, 13, 14, 8 | dihlspsnat 37139 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (◡𝐼‘(𝑁‘{𝑋})) ∈ (Atoms‘𝐾)) |
16 | 9, 10, 11, 15 | syl3anc 1476 | . . . . 5 ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑋})) ∈ (Atoms‘𝐾)) |
17 | dihprrn.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
18 | dihprrnlem2.yz | . . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 0 ) | |
19 | 5, 3, 6, 12, 13, 14, 8 | dihlspsnat 37139 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) → (◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾)) |
20 | 9, 17, 18, 19 | syl3anc 1476 | . . . . 5 ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾)) |
21 | 3, 4, 5, 6, 7, 8, 9, 16, 20 | dihjat 37229 | . . . 4 ⊢ (𝜑 → (𝐼‘((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌})))) = ((𝐼‘(◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(◡𝐼‘(𝑁‘{𝑌}))))) |
22 | 3, 6, 12, 14, 8 | dihlsprn 37137 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼) |
23 | 9, 10, 22 | syl2anc 565 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran 𝐼) |
24 | 3, 8 | dihcnvid2 37079 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (𝐼‘(◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋})) |
25 | 9, 23, 24 | syl2anc 565 | . . . . 5 ⊢ (𝜑 → (𝐼‘(◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋})) |
26 | 3, 6, 12, 14, 8 | dihlsprn 37137 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ ran 𝐼) |
27 | 9, 17, 26 | syl2anc 565 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ ran 𝐼) |
28 | 3, 8 | dihcnvid2 37079 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (𝐼‘(◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌})) |
29 | 9, 27, 28 | syl2anc 565 | . . . . 5 ⊢ (𝜑 → (𝐼‘(◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌})) |
30 | 25, 29 | oveq12d 6810 | . . . 4 ⊢ (𝜑 → ((𝐼‘(◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(◡𝐼‘(𝑁‘{𝑌})))) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌}))) |
31 | 3, 6, 9 | dvhlmod 36916 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
32 | 10 | snssd 4475 | . . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
33 | 17 | snssd 4475 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
34 | 12, 14, 7 | lsmsp2 19299 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌}))) |
35 | 31, 32, 33, 34 | syl3anc 1476 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌}))) |
36 | 21, 30, 35 | 3eqtrrd 2810 | . . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝐼‘((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌}))))) |
37 | 2, 36 | syl5eq 2817 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝐼‘((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌}))))) |
38 | 9 | simpld 476 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
39 | hllat 35168 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
40 | 38, 39 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat) |
41 | eqid 2771 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
42 | 41, 3, 8 | dihcnvcl 37077 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾)) |
43 | 9, 23, 42 | syl2anc 565 | . . . 4 ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾)) |
44 | 41, 3, 8 | dihcnvcl 37077 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾)) |
45 | 9, 27, 44 | syl2anc 565 | . . . 4 ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾)) |
46 | 41, 4 | latjcl 17258 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾) ∧ (◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾)) → ((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾)) |
47 | 40, 43, 45, 46 | syl3anc 1476 | . . 3 ⊢ (𝜑 → ((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾)) |
48 | 41, 3, 8 | dihcl 37076 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾)) → (𝐼‘((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌})))) ∈ ran 𝐼) |
49 | 9, 47, 48 | syl2anc 565 | . 2 ⊢ (𝜑 → (𝐼‘((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌})))) ∈ ran 𝐼) |
50 | 37, 49 | eqeltrd 2850 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∪ cun 3721 ⊆ wss 3723 {csn 4316 {cpr 4318 ◡ccnv 5248 ran crn 5250 ‘cfv 6031 (class class class)co 6792 Basecbs 16063 0gc0g 16307 joincjn 17151 Latclat 17252 LSSumclsm 18255 LModclmod 19072 LSpanclspn 19183 Atomscatm 35068 HLchlt 35155 LHypclh 35788 DVecHcdvh 36884 DIsoHcdih 37034 |
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 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-riotaBAD 34757 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 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 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-tpos 7503 df-undef 7550 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-n0 11494 df-z 11579 df-uz 11888 df-fz 12533 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-sca 16164 df-vsca 16165 df-0g 16309 df-preset 17135 df-poset 17153 df-plt 17165 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-p0 17246 df-p1 17247 df-lat 17253 df-clat 17315 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-grp 17632 df-minusg 17633 df-sbg 17634 df-subg 17798 df-cntz 17956 df-lsm 18257 df-cmn 18401 df-abl 18402 df-mgp 18697 df-ur 18709 df-ring 18756 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-dvr 18890 df-drng 18958 df-lmod 19074 df-lss 19142 df-lsp 19184 df-lvec 19315 df-lsatoms 34781 df-oposet 34981 df-ol 34983 df-oml 34984 df-covers 35071 df-ats 35072 df-atl 35103 df-cvlat 35127 df-hlat 35156 df-llines 35302 df-lplanes 35303 df-lvols 35304 df-lines 35305 df-psubsp 35307 df-pmap 35308 df-padd 35600 df-lhyp 35792 df-laut 35793 df-ldil 35908 df-ltrn 35909 df-trl 35964 df-tgrp 36548 df-tendo 36560 df-edring 36562 df-dveca 36808 df-disoa 36835 df-dvech 36885 df-dib 36945 df-dic 36979 df-dih 37035 df-doch 37154 df-djh 37201 |
This theorem is referenced by: dihprrn 37232 |
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