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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem8 | Structured version Visualization version GIF version |
Description: Part of proof of part 14 in [Baer] p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015.) |
Ref | Expression |
---|---|
hdmap14lem8.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap14lem8.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap14lem8.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap14lem8.q | ⊢ + = (+g‘𝑈) |
hdmap14lem8.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmap14lem8.o | ⊢ 0 = (0g‘𝑈) |
hdmap14lem8.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap14lem8.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmap14lem8.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmap14lem8.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap14lem8.d | ⊢ ✚ = (+g‘𝐶) |
hdmap14lem8.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hdmap14lem8.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hdmap14lem8.a | ⊢ 𝐴 = (Base‘𝑃) |
hdmap14lem8.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap14lem8.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap14lem8.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem8.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem8.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
hdmap14lem8.g | ⊢ (𝜑 → 𝐺 ∈ 𝐴) |
hdmap14lem8.i | ⊢ (𝜑 → 𝐼 ∈ 𝐴) |
hdmap14lem8.xx | ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) |
hdmap14lem8.yy | ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) |
hdmap14lem8.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
hdmap14lem8.j | ⊢ (𝜑 → 𝐽 ∈ 𝐴) |
hdmap14lem8.xy | ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌)))) |
Ref | Expression |
---|---|
hdmap14lem8 | ⊢ (𝜑 → ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem8.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap14lem8.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
3 | hdmap14lem8.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | lcdlmod 41060 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | hdmap14lem8.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐴) | |
6 | hdmap14lem8.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | hdmap14lem8.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
8 | eqid 2728 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | hdmap14lem8.s | . . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
10 | hdmap14lem8.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
11 | 10 | eldifad 3957 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
12 | 1, 6, 7, 2, 8, 9, 3, 11 | hdmapcl 41298 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶)) |
13 | hdmap14lem8.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
14 | 13 | eldifad 3957 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
15 | 1, 6, 7, 2, 8, 9, 3, 14 | hdmapcl 41298 | . . 3 ⊢ (𝜑 → (𝑆‘𝑌) ∈ (Base‘𝐶)) |
16 | hdmap14lem8.d | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
17 | hdmap14lem8.p | . . . 4 ⊢ 𝑃 = (Scalar‘𝐶) | |
18 | hdmap14lem8.e | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
19 | hdmap14lem8.a | . . . 4 ⊢ 𝐴 = (Base‘𝑃) | |
20 | 8, 16, 17, 18, 19 | lmodvsdi 20762 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ (𝐽 ∈ 𝐴 ∧ (𝑆‘𝑋) ∈ (Base‘𝐶) ∧ (𝑆‘𝑌) ∈ (Base‘𝐶))) → (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) = ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌)))) |
21 | 4, 5, 12, 15, 20 | syl13anc 1370 | . 2 ⊢ (𝜑 → (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) = ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌)))) |
22 | hdmap14lem8.q | . . . . 5 ⊢ + = (+g‘𝑈) | |
23 | 1, 6, 7, 22, 2, 16, 9, 3, 11, 14 | hdmapadd 41311 | . . . 4 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) |
24 | 23 | oveq2d 7431 | . . 3 ⊢ (𝜑 → (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))) = (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))) |
25 | hdmap14lem8.xy | . . . 4 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌)))) | |
26 | 1, 6, 3 | dvhlmod 40578 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
27 | hdmap14lem8.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
28 | hdmap14lem8.r | . . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑈) | |
29 | hdmap14lem8.t | . . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑈) | |
30 | hdmap14lem8.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅) | |
31 | 7, 22, 28, 29, 30 | lmodvsdi 20762 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐹 · (𝑋 + 𝑌)) = ((𝐹 · 𝑋) + (𝐹 · 𝑌))) |
32 | 26, 27, 11, 14, 31 | syl13anc 1370 | . . . . . 6 ⊢ (𝜑 → (𝐹 · (𝑋 + 𝑌)) = ((𝐹 · 𝑋) + (𝐹 · 𝑌))) |
33 | 32 | fveq2d 6896 | . . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑆‘((𝐹 · 𝑋) + (𝐹 · 𝑌)))) |
34 | 7, 28, 29, 30 | lmodvscl 20755 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉) |
35 | 26, 27, 11, 34 | syl3anc 1369 | . . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉) |
36 | 7, 28, 29, 30 | lmodvscl 20755 | . . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → (𝐹 · 𝑌) ∈ 𝑉) |
37 | 26, 27, 14, 36 | syl3anc 1369 | . . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑌) ∈ 𝑉) |
38 | 1, 6, 7, 22, 2, 16, 9, 3, 35, 37 | hdmapadd 41311 | . . . . 5 ⊢ (𝜑 → (𝑆‘((𝐹 · 𝑋) + (𝐹 · 𝑌))) = ((𝑆‘(𝐹 · 𝑋)) ✚ (𝑆‘(𝐹 · 𝑌)))) |
39 | hdmap14lem8.xx | . . . . . 6 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋))) | |
40 | hdmap14lem8.yy | . . . . . 6 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌))) | |
41 | 39, 40 | oveq12d 7433 | . . . . 5 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) ✚ (𝑆‘(𝐹 · 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌)))) |
42 | 33, 38, 41 | 3eqtrd 2772 | . . . 4 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌)))) |
43 | 25, 42 | eqtr3d 2770 | . . 3 ⊢ (𝜑 → (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌)))) |
44 | 24, 43 | eqtr3d 2770 | . 2 ⊢ (𝜑 → (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌)))) |
45 | 21, 44 | eqtr3d 2770 | 1 ⊢ (𝜑 → ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∖ cdif 3942 {csn 4625 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 +gcplusg 17227 Scalarcsca 17230 ·𝑠 cvsca 17231 0gc0g 17415 LModclmod 20737 LSpanclspn 20849 HLchlt 38817 LHypclh 39452 DVecHcdvh 40546 LCDualclcd 41054 HDMapchdma 41260 |
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 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-riotaBAD 38420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-tpos 8226 df-undef 8273 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17417 df-mre 17560 df-mrc 17561 df-acs 17563 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18887 df-minusg 18888 df-sbg 18889 df-subg 19072 df-cntz 19262 df-oppg 19291 df-lsm 19585 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-invr 20321 df-dvr 20334 df-drng 20620 df-lmod 20739 df-lss 20810 df-lsp 20850 df-lvec 20982 df-lsatoms 38443 df-lshyp 38444 df-lcv 38486 df-lfl 38525 df-lkr 38553 df-ldual 38591 df-oposet 38643 df-ol 38645 df-oml 38646 df-covers 38733 df-ats 38734 df-atl 38765 df-cvlat 38789 df-hlat 38818 df-llines 38966 df-lplanes 38967 df-lvols 38968 df-lines 38969 df-psubsp 38971 df-pmap 38972 df-padd 39264 df-lhyp 39456 df-laut 39457 df-ldil 39572 df-ltrn 39573 df-trl 39627 df-tgrp 40211 df-tendo 40223 df-edring 40225 df-dveca 40471 df-disoa 40497 df-dvech 40547 df-dib 40607 df-dic 40641 df-dih 40697 df-doch 40816 df-djh 40863 df-lcdual 41055 df-mapd 41093 df-hvmap 41225 df-hdmap1 41261 df-hdmap 41262 |
This theorem is referenced by: hdmap14lem9 41344 |
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