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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 2726 | . . . . . 6 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
5 | eqid 2726 | . . . . . 6 ⊢ (invg‘𝑈) = (invg‘𝑈) | |
6 | hdmap12c.m | . . . . . 6 ⊢ − = (-g‘𝑈) | |
7 | 3, 4, 5, 6 | grpsubval 18975 | . . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) = (𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌))) |
8 | 1, 2, 7 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌))) |
9 | 8 | fveq2d 6897 | . . 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 2726 | . . . 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 40822 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
17 | 3, 5 | lmodvnegcl 20875 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → ((invg‘𝑈)‘𝑌) ∈ 𝑉) |
18 | 16, 2, 17 | syl2anc 582 | . . . 4 ⊢ (𝜑 → ((invg‘𝑈)‘𝑌) ∈ 𝑉) |
19 | 10, 11, 3, 4, 12, 13, 14, 15, 1, 18 | hdmapadd 41555 | . . 3 ⊢ (𝜑 → (𝑆‘(𝑋(+g‘𝑈)((invg‘𝑈)‘𝑌))) = ((𝑆‘𝑋)(+g‘𝐶)(𝑆‘((invg‘𝑈)‘𝑌)))) |
20 | eqid 2726 | . . . . 5 ⊢ (invg‘𝐶) = (invg‘𝐶) | |
21 | 10, 11, 3, 5, 12, 20, 14, 15, 2 | hdmapneg 41558 | . . . 4 ⊢ (𝜑 → (𝑆‘((invg‘𝑈)‘𝑌)) = ((invg‘𝐶)‘(𝑆‘𝑌))) |
22 | 21 | oveq2d 7432 | . . 3 ⊢ (𝜑 → ((𝑆‘𝑋)(+g‘𝐶)(𝑆‘((invg‘𝑈)‘𝑌))) = ((𝑆‘𝑋)(+g‘𝐶)((invg‘𝐶)‘(𝑆‘𝑌)))) |
23 | 9, 19, 22 | 3eqtrd 2770 | . 2 ⊢ (𝜑 → (𝑆‘(𝑋 − 𝑌)) = ((𝑆‘𝑋)(+g‘𝐶)((invg‘𝐶)‘(𝑆‘𝑌)))) |
24 | eqid 2726 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
25 | 10, 11, 3, 12, 24, 14, 15, 1 | hdmapcl 41542 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶)) |
26 | 10, 11, 3, 12, 24, 14, 15, 2 | hdmapcl 41542 | . . 3 ⊢ (𝜑 → (𝑆‘𝑌) ∈ (Base‘𝐶)) |
27 | hdmap12c.n | . . . 4 ⊢ 𝑁 = (-g‘𝐶) | |
28 | 24, 13, 20, 27 | grpsubval 18975 | . . 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 582 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋)𝑁(𝑆‘𝑌)) = ((𝑆‘𝑋)(+g‘𝐶)((invg‘𝐶)‘(𝑆‘𝑌)))) |
30 | 23, 29 | eqtr4d 2769 | 1 ⊢ (𝜑 → (𝑆‘(𝑋 − 𝑌)) = ((𝑆‘𝑋)𝑁(𝑆‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 +gcplusg 17261 invgcminusg 18924 -gcsg 18925 LModclmod 20832 HLchlt 39061 LHypclh 39696 DVecHcdvh 40790 LCDualclcd 41298 HDMapchdma 41504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-riotaBAD 38664 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-tpos 8233 df-undef 8280 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-0g 17451 df-mre 17594 df-mrc 17595 df-acs 17597 df-proset 18315 df-poset 18333 df-plt 18350 df-lub 18366 df-glb 18367 df-join 18368 df-meet 18369 df-p0 18445 df-p1 18446 df-lat 18452 df-clat 18519 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19113 df-cntz 19307 df-oppg 19336 df-lsm 19630 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-nzr 20491 df-rlreg 20668 df-domn 20669 df-drng 20705 df-lmod 20834 df-lss 20905 df-lsp 20945 df-lvec 21077 df-lsatoms 38687 df-lshyp 38688 df-lcv 38730 df-lfl 38769 df-lkr 38797 df-ldual 38835 df-oposet 38887 df-ol 38889 df-oml 38890 df-covers 38977 df-ats 38978 df-atl 39009 df-cvlat 39033 df-hlat 39062 df-llines 39210 df-lplanes 39211 df-lvols 39212 df-lines 39213 df-psubsp 39215 df-pmap 39216 df-padd 39508 df-lhyp 39700 df-laut 39701 df-ldil 39816 df-ltrn 39817 df-trl 39871 df-tgrp 40455 df-tendo 40467 df-edring 40469 df-dveca 40715 df-disoa 40741 df-dvech 40791 df-dib 40851 df-dic 40885 df-dih 40941 df-doch 41060 df-djh 41107 df-lcdual 41299 df-mapd 41337 df-hvmap 41469 df-hdmap1 41505 df-hdmap 41506 |
This theorem is referenced by: hdmap11 41560 hdmapinvlem3 41632 |
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