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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapsub | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 5, (a-b)S = aS-bS in their notation (S = sigma). (Contributed by NM, 26-May-2015.) |
Ref | Expression |
---|---|
hdmap12c.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap12c.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap12c.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap12c.m | ⊢ − = (-g‘𝑈) |
hdmap12c.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap12c.n | ⊢ 𝑁 = (-g‘𝐶) |
hdmap12c.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap12c.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap12c.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
hdmap12c.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
hdmapsub | ⊢ (𝜑 → (𝑆‘(𝑋 − 𝑌)) = ((𝑆‘𝑋)𝑁(𝑆‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap12c.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | hdmap12c.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
3 | hdmap12c.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑈) | |
4 | eqid 2771 | . . . . . 6 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
5 | eqid 2771 | . . . . . 6 ⊢ (invg‘𝑈) = (invg‘𝑈) | |
6 | hdmap12c.m | . . . . . 6 ⊢ − = (-g‘𝑈) | |
7 | 3, 4, 5, 6 | grpsubval 17672 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) = (𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌))) |
8 | 1, 2, 7 | syl2anc 565 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌))) |
9 | 8 | fveq2d 6336 | . . 3 ⊢ (𝜑 → (𝑆‘(𝑋 − 𝑌)) = (𝑆‘(𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌)))) |
10 | hdmap12c.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | hdmap12c.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
12 | hdmap12c.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
13 | eqid 2771 | . . . 4 ⊢ (+g‘𝐶) = (+g‘𝐶) | |
14 | hdmap12c.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
15 | hdmap12c.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
16 | 10, 11, 15 | dvhlmod 36916 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
17 | 3, 5 | lmodvnegcl 19113 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((invg‘𝑈)‘𝑌) ∈ 𝑉) |
18 | 16, 2, 17 | syl2anc 565 | . . . 4 ⊢ (𝜑 → ((invg‘𝑈)‘𝑌) ∈ 𝑉) |
19 | 10, 11, 3, 4, 12, 13, 14, 15, 1, 18 | hdmapadd 37649 | . . 3 ⊢ (𝜑 → (𝑆‘(𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌))) = ((𝑆‘𝑋)(+g‘𝐶)(𝑆‘((invg‘𝑈)‘𝑌)))) |
20 | eqid 2771 | . . . . 5 ⊢ (invg‘𝐶) = (invg‘𝐶) | |
21 | 10, 11, 3, 5, 12, 20, 14, 15, 2 | hdmapneg 37652 | . . . 4 ⊢ (𝜑 → (𝑆‘((invg‘𝑈)‘𝑌)) = ((invg‘𝐶)‘(𝑆‘𝑌))) |
22 | 21 | oveq2d 6808 | . . 3 ⊢ (𝜑 → ((𝑆‘𝑋)(+g‘𝐶)(𝑆‘((invg‘𝑈)‘𝑌))) = ((𝑆‘𝑋)(+g‘𝐶)((invg‘𝐶)‘(𝑆‘𝑌)))) |
23 | 9, 19, 22 | 3eqtrd 2809 | . 2 ⊢ (𝜑 → (𝑆‘(𝑋 − 𝑌)) = ((𝑆‘𝑋)(+g‘𝐶)((invg‘𝐶)‘(𝑆‘𝑌)))) |
24 | eqid 2771 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
25 | 10, 11, 3, 12, 24, 14, 15, 1 | hdmapcl 37636 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶)) |
26 | 10, 11, 3, 12, 24, 14, 15, 2 | hdmapcl 37636 | . . 3 ⊢ (𝜑 → (𝑆‘𝑌) ∈ (Base‘𝐶)) |
27 | hdmap12c.n | . . . 4 ⊢ 𝑁 = (-g‘𝐶) | |
28 | 24, 13, 20, 27 | grpsubval 17672 | . . 3 ⊢ (((𝑆‘𝑋) ∈ (Base‘𝐶) ∧ (𝑆‘𝑌) ∈ (Base‘𝐶)) → ((𝑆‘𝑋)𝑁(𝑆‘𝑌)) = ((𝑆‘𝑋)(+g‘𝐶)((invg‘𝐶)‘(𝑆‘𝑌)))) |
29 | 25, 26, 28 | syl2anc 565 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋)𝑁(𝑆‘𝑌)) = ((𝑆‘𝑋)(+g‘𝐶)((invg‘𝐶)‘(𝑆‘𝑌)))) |
30 | 23, 29 | eqtr4d 2808 | 1 ⊢ (𝜑 → (𝑆‘(𝑋 − 𝑌)) = ((𝑆‘𝑋)𝑁(𝑆‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ‘cfv 6031 (class class class)co 6792 Basecbs 16063 +gcplusg 16148 invgcminusg 17630 -gcsg 17631 LModclmod 19072 HLchlt 35155 LHypclh 35788 DVecHcdvh 36884 LCDualclcd 37392 HDMapchdma 37598 |
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-ot 4325 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-of 7043 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-mre 16453 df-mrc 16454 df-acs 16456 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-oppg 17982 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-lshyp 34782 df-lcv 34824 df-lfl 34863 df-lkr 34891 df-ldual 34929 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 df-lcdual 37393 df-mapd 37431 df-hvmap 37563 df-hdmap1 37599 df-hdmap 37600 |
This theorem is referenced by: hdmap11 37654 hdmapinvlem3 37726 |
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