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Theorem hdmap1eq4N 40672
Description: Convert mapdheq4 40598 to use HDMap1 function. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hdmap1eq2.h 𝐻 = (LHypβ€˜πΎ)
hdmap1eq2.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hdmap1eq2.v 𝑉 = (Baseβ€˜π‘ˆ)
hdmap1eq2.o 0 = (0gβ€˜π‘ˆ)
hdmap1eq2.n 𝑁 = (LSpanβ€˜π‘ˆ)
hdmap1eq2.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
hdmap1eq2.d 𝐷 = (Baseβ€˜πΆ)
hdmap1eq2.l 𝐿 = (LSpanβ€˜πΆ)
hdmap1eq2.m 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
hdmap1eq2.i 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
hdmap1eq2.k (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
hdmap1eq2.f (πœ‘ β†’ 𝐹 ∈ 𝐷)
hdmap1eq2.mn (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (πΏβ€˜{𝐹}))
hdmap1eq4.x (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
hdmap1eq4.y (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
hdmap1eq4.z (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
hdmap1eq4.ne (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
hdmap1eq4.xn (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
hdmap1eq4.eg (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
hdmap1eq4.ee (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐡)
Assertion
Ref Expression
hdmap1eq4N (πœ‘ β†’ (πΌβ€˜βŸ¨π‘Œ, 𝐺, π‘βŸ©) = 𝐡)

Proof of Theorem hdmap1eq4N
Dummy variables π‘₯ β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmap1eq2.h . . 3 𝐻 = (LHypβ€˜πΎ)
2 hdmap1eq2.u . . 3 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
3 hdmap1eq2.v . . 3 𝑉 = (Baseβ€˜π‘ˆ)
4 eqid 2732 . . 3 (-gβ€˜π‘ˆ) = (-gβ€˜π‘ˆ)
5 hdmap1eq2.o . . 3 0 = (0gβ€˜π‘ˆ)
6 hdmap1eq2.n . . 3 𝑁 = (LSpanβ€˜π‘ˆ)
7 hdmap1eq2.c . . 3 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
8 hdmap1eq2.d . . 3 𝐷 = (Baseβ€˜πΆ)
9 eqid 2732 . . 3 (-gβ€˜πΆ) = (-gβ€˜πΆ)
10 eqid 2732 . . 3 (0gβ€˜πΆ) = (0gβ€˜πΆ)
11 hdmap1eq2.l . . 3 𝐿 = (LSpanβ€˜πΆ)
12 hdmap1eq2.m . . 3 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
13 hdmap1eq2.i . . 3 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
14 hdmap1eq2.k . . 3 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
15 hdmap1eq4.y . . 3 (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
16 hdmap1eq4.eg . . . 4 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
17 hdmap1eq2.f . . . . 5 (πœ‘ β†’ 𝐹 ∈ 𝐷)
18 hdmap1eq2.mn . . . . 5 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (πΏβ€˜{𝐹}))
191, 2, 14dvhlvec 39975 . . . . . . 7 (πœ‘ β†’ π‘ˆ ∈ LVec)
20 hdmap1eq4.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
2120eldifad 3960 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ 𝑉)
2215eldifad 3960 . . . . . . 7 (πœ‘ β†’ π‘Œ ∈ 𝑉)
23 hdmap1eq4.z . . . . . . . 8 (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
2423eldifad 3960 . . . . . . 7 (πœ‘ β†’ 𝑍 ∈ 𝑉)
25 hdmap1eq4.xn . . . . . . 7 (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
263, 6, 19, 21, 22, 24, 25lspindpi 20744 . . . . . 6 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}) ∧ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍})))
2726simpld 495 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
281, 2, 3, 5, 6, 7, 8, 11, 12, 13, 14, 17, 18, 27, 20, 22hdmap1cl 40670 . . . 4 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) ∈ 𝐷)
2916, 28eqeltrrd 2834 . . 3 (πœ‘ β†’ 𝐺 ∈ 𝐷)
30 eqid 2732 . . 3 (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , (0gβ€˜πΆ), (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (πΏβ€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘ˆ)(2nd β€˜π‘₯))})) = (πΏβ€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜πΆ)β„Ž)}))))) = (π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , (0gβ€˜πΆ), (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (πΏβ€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘ˆ)(2nd β€˜π‘₯))})) = (πΏβ€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜πΆ)β„Ž)})))))
311, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 29, 24, 30hdmap1valc 40669 . 2 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘Œ, 𝐺, π‘βŸ©) = ((π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , (0gβ€˜πΆ), (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (πΏβ€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘ˆ)(2nd β€˜π‘₯))})) = (πΏβ€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜πΆ)β„Ž)})))))β€˜βŸ¨π‘Œ, 𝐺, π‘βŸ©))
32 hdmap1eq4.ne . . 3 (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
331, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 17, 22, 30hdmap1valc 40669 . . . 4 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = ((π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , (0gβ€˜πΆ), (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (πΏβ€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘ˆ)(2nd β€˜π‘₯))})) = (πΏβ€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜πΆ)β„Ž)})))))β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))
3433, 16eqtr3d 2774 . . 3 (πœ‘ β†’ ((π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , (0gβ€˜πΆ), (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (πΏβ€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘ˆ)(2nd β€˜π‘₯))})) = (πΏβ€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜πΆ)β„Ž)})))))β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 𝐺)
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 20, 17, 24, 30hdmap1valc 40669 . . . 4 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = ((π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , (0gβ€˜πΆ), (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (πΏβ€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘ˆ)(2nd β€˜π‘₯))})) = (πΏβ€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜πΆ)β„Ž)})))))β€˜βŸ¨π‘‹, 𝐹, π‘βŸ©))
36 hdmap1eq4.ee . . . 4 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐡)
3735, 36eqtr3d 2774 . . 3 (πœ‘ β†’ ((π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , (0gβ€˜πΆ), (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (πΏβ€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘ˆ)(2nd β€˜π‘₯))})) = (πΏβ€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜πΆ)β„Ž)})))))β€˜βŸ¨π‘‹, 𝐹, π‘βŸ©) = 𝐡)
3810, 30, 1, 12, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 17, 18, 20, 15, 23, 25, 32, 34, 37mapdheq4 40598 . 2 (πœ‘ β†’ ((π‘₯ ∈ V ↦ if((2nd β€˜π‘₯) = 0 , (0gβ€˜πΆ), (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (πΏβ€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯))(-gβ€˜π‘ˆ)(2nd β€˜π‘₯))})) = (πΏβ€˜{((2nd β€˜(1st β€˜π‘₯))(-gβ€˜πΆ)β„Ž)})))))β€˜βŸ¨π‘Œ, 𝐺, π‘βŸ©) = 𝐡)
3931, 38eqtrd 2772 1 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘Œ, 𝐺, π‘βŸ©) = 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  Vcvv 3474   βˆ– cdif 3945  ifcif 4528  {csn 4628  {cpr 4630  βŸ¨cotp 4636   ↦ cmpt 5231  β€˜cfv 6543  β„©crio 7363  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  Basecbs 17143  0gc0g 17384  -gcsg 18820  LSpanclspn 20581  HLchlt 38215  LHypclh 38850  DVecHcdvh 39944  LCDualclcd 40452  mapdcmpd 40490  HDMap1chdma1 40657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-riotaBAD 37818
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7669  df-om 7855  df-1st 7974  df-2nd 7975  df-tpos 8210  df-undef 8257  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-0g 17386  df-mre 17529  df-mrc 17530  df-acs 17532  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-submnd 18671  df-grp 18821  df-minusg 18822  df-sbg 18823  df-subg 19002  df-cntz 19180  df-oppg 19209  df-lsm 19503  df-cmn 19649  df-abl 19650  df-mgp 19987  df-ur 20004  df-ring 20057  df-oppr 20149  df-dvdsr 20170  df-unit 20171  df-invr 20201  df-dvr 20214  df-drng 20358  df-lmod 20472  df-lss 20542  df-lsp 20582  df-lvec 20713  df-lsatoms 37841  df-lshyp 37842  df-lcv 37884  df-lfl 37923  df-lkr 37951  df-ldual 37989  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-llines 38364  df-lplanes 38365  df-lvols 38366  df-lines 38367  df-psubsp 38369  df-pmap 38370  df-padd 38662  df-lhyp 38854  df-laut 38855  df-ldil 38970  df-ltrn 38971  df-trl 39025  df-tgrp 39609  df-tendo 39621  df-edring 39623  df-dveca 39869  df-disoa 39895  df-dvech 39945  df-dib 40005  df-dic 40039  df-dih 40095  df-doch 40214  df-djh 40261  df-lcdual 40453  df-mapd 40491  df-hdmap1 40659
This theorem is referenced by:  hdmapval3lemN  40703
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