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Theorem hdmap14lem6 39895
Description: Case where 𝐹 is zero. (Contributed by NM, 1-Jun-2015.)
Hypotheses
Ref Expression
hdmap14lem1.h 𝐻 = (LHyp‘𝐾)
hdmap14lem1.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap14lem1.v 𝑉 = (Base‘𝑈)
hdmap14lem1.t · = ( ·𝑠𝑈)
hdmap14lem3.o 0 = (0g𝑈)
hdmap14lem1.r 𝑅 = (Scalar‘𝑈)
hdmap14lem1.b 𝐵 = (Base‘𝑅)
hdmap14lem1.z 𝑍 = (0g𝑅)
hdmap14lem1.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap14lem2.e = ( ·𝑠𝐶)
hdmap14lem1.l 𝐿 = (LSpan‘𝐶)
hdmap14lem2.p 𝑃 = (Scalar‘𝐶)
hdmap14lem2.a 𝐴 = (Base‘𝑃)
hdmap14lem2.q 𝑄 = (0g𝑃)
hdmap14lem1.s 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap14lem1.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
hdmap14lem3.x (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap14lem6.f (𝜑𝐹 = 𝑍)
Assertion
Ref Expression
hdmap14lem6 (𝜑 → ∃!𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)))
Distinct variable groups:   𝐴,𝑔   𝐶,𝑔   ,𝑔   𝑄,𝑔   𝑆,𝑔   𝑔,𝑋   𝜑,𝑔
Allowed substitution hints:   𝐵(𝑔)   𝑃(𝑔)   𝑅(𝑔)   · (𝑔)   𝑈(𝑔)   𝐹(𝑔)   𝐻(𝑔)   𝐾(𝑔)   𝐿(𝑔)   𝑉(𝑔)   𝑊(𝑔)   0 (𝑔)   𝑍(𝑔)

Proof of Theorem hdmap14lem6
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 hdmap14lem1.h . . . . . 6 𝐻 = (LHyp‘𝐾)
2 hdmap14lem1.c . . . . . 6 𝐶 = ((LCDual‘𝐾)‘𝑊)
3 hdmap14lem1.k . . . . . 6 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
41, 2, 3lcdlmod 39614 . . . . 5 (𝜑𝐶 ∈ LMod)
5 hdmap14lem2.p . . . . . 6 𝑃 = (Scalar‘𝐶)
6 hdmap14lem2.a . . . . . 6 𝐴 = (Base‘𝑃)
7 hdmap14lem2.q . . . . . 6 𝑄 = (0g𝑃)
85, 6, 7lmod0cl 20159 . . . . 5 (𝐶 ∈ LMod → 𝑄𝐴)
94, 8syl 17 . . . 4 (𝜑𝑄𝐴)
10 hdmap14lem1.u . . . . . . 7 𝑈 = ((DVecH‘𝐾)‘𝑊)
11 hdmap14lem1.v . . . . . . 7 𝑉 = (Base‘𝑈)
12 eqid 2738 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
13 hdmap14lem1.s . . . . . . 7 𝑆 = ((HDMap‘𝐾)‘𝑊)
14 hdmap14lem3.x . . . . . . . 8 (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
1514eldifad 3898 . . . . . . 7 (𝜑𝑋𝑉)
161, 10, 11, 2, 12, 13, 3, 15hdmapcl 39852 . . . . . 6 (𝜑 → (𝑆𝑋) ∈ (Base‘𝐶))
17 hdmap14lem2.e . . . . . . 7 = ( ·𝑠𝐶)
18 eqid 2738 . . . . . . 7 (0g𝐶) = (0g𝐶)
1912, 5, 17, 7, 18lmod0vs 20166 . . . . . 6 ((𝐶 ∈ LMod ∧ (𝑆𝑋) ∈ (Base‘𝐶)) → (𝑄 (𝑆𝑋)) = (0g𝐶))
204, 16, 19syl2anc 584 . . . . 5 (𝜑 → (𝑄 (𝑆𝑋)) = (0g𝐶))
2120eqcomd 2744 . . . 4 (𝜑 → (0g𝐶) = (𝑄 (𝑆𝑋)))
22 oveq1 7274 . . . . 5 (𝑔 = 𝑄 → (𝑔 (𝑆𝑋)) = (𝑄 (𝑆𝑋)))
2322rspceeqv 3574 . . . 4 ((𝑄𝐴 ∧ (0g𝐶) = (𝑄 (𝑆𝑋))) → ∃𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋)))
249, 21, 23syl2anc 584 . . 3 (𝜑 → ∃𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋)))
25 hdmap14lem3.o . . . . . . . . . . 11 0 = (0g𝑈)
261, 10, 11, 25, 2, 18, 12, 13, 3, 14hdmapnzcl 39867 . . . . . . . . . 10 (𝜑 → (𝑆𝑋) ∈ ((Base‘𝐶) ∖ {(0g𝐶)}))
27 eldifsni 4723 . . . . . . . . . 10 ((𝑆𝑋) ∈ ((Base‘𝐶) ∖ {(0g𝐶)}) → (𝑆𝑋) ≠ (0g𝐶))
2826, 27syl 17 . . . . . . . . 9 (𝜑 → (𝑆𝑋) ≠ (0g𝐶))
2928neneqd 2948 . . . . . . . 8 (𝜑 → ¬ (𝑆𝑋) = (0g𝐶))
30293ad2ant1 1132 . . . . . . 7 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ¬ (𝑆𝑋) = (0g𝐶))
31 simp3l 1200 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (0g𝐶) = (𝑔 (𝑆𝑋)))
3231eqcomd 2744 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (𝑔 (𝑆𝑋)) = (0g𝐶))
331, 2, 3lcdlvec 39613 . . . . . . . . . . . 12 (𝜑𝐶 ∈ LVec)
34333ad2ant1 1132 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → 𝐶 ∈ LVec)
35 simp2l 1198 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → 𝑔𝐴)
36163ad2ant1 1132 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (𝑆𝑋) ∈ (Base‘𝐶))
3712, 17, 5, 6, 7, 18, 34, 35, 36lvecvs0or 20380 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ((𝑔 (𝑆𝑋)) = (0g𝐶) ↔ (𝑔 = 𝑄 ∨ (𝑆𝑋) = (0g𝐶))))
3832, 37mpbid 231 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (𝑔 = 𝑄 ∨ (𝑆𝑋) = (0g𝐶)))
3938orcomd 868 . . . . . . . 8 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ((𝑆𝑋) = (0g𝐶) ∨ 𝑔 = 𝑄))
4039ord 861 . . . . . . 7 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (¬ (𝑆𝑋) = (0g𝐶) → 𝑔 = 𝑄))
4130, 40mpd 15 . . . . . 6 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → 𝑔 = 𝑄)
42 simp3r 1201 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (0g𝐶) = ( (𝑆𝑋)))
4342eqcomd 2744 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ( (𝑆𝑋)) = (0g𝐶))
44 simp2r 1199 . . . . . . . . . . 11 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → 𝐴)
4512, 17, 5, 6, 7, 18, 34, 44, 36lvecvs0or 20380 . . . . . . . . . 10 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (( (𝑆𝑋)) = (0g𝐶) ↔ ( = 𝑄 ∨ (𝑆𝑋) = (0g𝐶))))
4643, 45mpbid 231 . . . . . . . . 9 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ( = 𝑄 ∨ (𝑆𝑋) = (0g𝐶)))
4746orcomd 868 . . . . . . . 8 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → ((𝑆𝑋) = (0g𝐶) ∨ = 𝑄))
4847ord 861 . . . . . . 7 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → (¬ (𝑆𝑋) = (0g𝐶) → = 𝑄))
4930, 48mpd 15 . . . . . 6 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → = 𝑄)
5041, 49eqtr4d 2781 . . . . 5 ((𝜑 ∧ (𝑔𝐴𝐴) ∧ ((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋)))) → 𝑔 = )
51503exp 1118 . . . 4 (𝜑 → ((𝑔𝐴𝐴) → (((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋))) → 𝑔 = )))
5251ralrimivv 3114 . . 3 (𝜑 → ∀𝑔𝐴𝐴 (((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋))) → 𝑔 = ))
53 oveq1 7274 . . . . 5 (𝑔 = → (𝑔 (𝑆𝑋)) = ( (𝑆𝑋)))
5453eqeq2d 2749 . . . 4 (𝑔 = → ((0g𝐶) = (𝑔 (𝑆𝑋)) ↔ (0g𝐶) = ( (𝑆𝑋))))
5554reu4 3665 . . 3 (∃!𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋)) ↔ (∃𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋)) ∧ ∀𝑔𝐴𝐴 (((0g𝐶) = (𝑔 (𝑆𝑋)) ∧ (0g𝐶) = ( (𝑆𝑋))) → 𝑔 = )))
5624, 52, 55sylanbrc 583 . 2 (𝜑 → ∃!𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋)))
57 hdmap14lem6.f . . . . . . . 8 (𝜑𝐹 = 𝑍)
5857oveq1d 7282 . . . . . . 7 (𝜑 → (𝐹 · 𝑋) = (𝑍 · 𝑋))
591, 10, 3dvhlmod 39132 . . . . . . . 8 (𝜑𝑈 ∈ LMod)
60 hdmap14lem1.r . . . . . . . . 9 𝑅 = (Scalar‘𝑈)
61 hdmap14lem1.t . . . . . . . . 9 · = ( ·𝑠𝑈)
62 hdmap14lem1.z . . . . . . . . 9 𝑍 = (0g𝑅)
6311, 60, 61, 62, 25lmod0vs 20166 . . . . . . . 8 ((𝑈 ∈ LMod ∧ 𝑋𝑉) → (𝑍 · 𝑋) = 0 )
6459, 15, 63syl2anc 584 . . . . . . 7 (𝜑 → (𝑍 · 𝑋) = 0 )
6558, 64eqtrd 2778 . . . . . 6 (𝜑 → (𝐹 · 𝑋) = 0 )
6665fveq2d 6770 . . . . 5 (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝑆0 ))
671, 10, 25, 2, 18, 13, 3hdmapval0 39855 . . . . 5 (𝜑 → (𝑆0 ) = (0g𝐶))
6866, 67eqtrd 2778 . . . 4 (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (0g𝐶))
6968eqeq1d 2740 . . 3 (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)) ↔ (0g𝐶) = (𝑔 (𝑆𝑋))))
7069reubidv 3321 . 2 (𝜑 → (∃!𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)) ↔ ∃!𝑔𝐴 (0g𝐶) = (𝑔 (𝑆𝑋))))
7156, 70mpbird 256 1 (𝜑 → ∃!𝑔𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 (𝑆𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 844  w3a 1086   = wceq 1539  wcel 2106  wne 2943  wral 3064  wrex 3065  ∃!wreu 3066  cdif 3883  {csn 4561  cfv 6426  (class class class)co 7267  Basecbs 16922  Scalarcsca 16975   ·𝑠 cvsca 16976  0gc0g 17160  LModclmod 20133  LSpanclspn 20243  LVecclvec 20374  HLchlt 37372  LHypclh 38006  DVecHcdvh 39100  LCDualclcd 39608  HDMapchdma 39814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578  ax-cnex 10937  ax-resscn 10938  ax-1cn 10939  ax-icn 10940  ax-addcl 10941  ax-addrcl 10942  ax-mulcl 10943  ax-mulrcl 10944  ax-mulcom 10945  ax-addass 10946  ax-mulass 10947  ax-distr 10948  ax-i2m1 10949  ax-1ne0 10950  ax-1rid 10951  ax-rnegex 10952  ax-rrecex 10953  ax-cnre 10954  ax-pre-lttri 10955  ax-pre-lttrn 10956  ax-pre-ltadd 10957  ax-pre-mulgt0 10958  ax-riotaBAD 36975
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-pred 6195  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-riota 7224  df-ov 7270  df-oprab 7271  df-mpo 7272  df-of 7523  df-om 7703  df-1st 7820  df-2nd 7821  df-tpos 8029  df-undef 8076  df-frecs 8084  df-wrecs 8115  df-recs 8189  df-rdg 8228  df-1o 8284  df-er 8485  df-map 8604  df-en 8721  df-dom 8722  df-sdom 8723  df-fin 8724  df-pnf 11021  df-mnf 11022  df-xr 11023  df-ltxr 11024  df-le 11025  df-sub 11217  df-neg 11218  df-nn 11984  df-2 12046  df-3 12047  df-4 12048  df-5 12049  df-6 12050  df-n0 12244  df-z 12330  df-uz 12593  df-fz 13250  df-struct 16858  df-sets 16875  df-slot 16893  df-ndx 16905  df-base 16923  df-ress 16952  df-plusg 16985  df-mulr 16986  df-sca 16988  df-vsca 16989  df-0g 17162  df-mre 17305  df-mrc 17306  df-acs 17308  df-proset 18023  df-poset 18041  df-plt 18058  df-lub 18074  df-glb 18075  df-join 18076  df-meet 18077  df-p0 18153  df-p1 18154  df-lat 18160  df-clat 18227  df-mgm 18336  df-sgrp 18385  df-mnd 18396  df-submnd 18441  df-grp 18590  df-minusg 18591  df-sbg 18592  df-subg 18762  df-cntz 18933  df-oppg 18960  df-lsm 19251  df-cmn 19398  df-abl 19399  df-mgp 19731  df-ur 19748  df-ring 19795  df-oppr 19872  df-dvdsr 19893  df-unit 19894  df-invr 19924  df-dvr 19935  df-drng 20003  df-lmod 20135  df-lss 20204  df-lsp 20244  df-lvec 20375  df-lsatoms 36998  df-lshyp 36999  df-lcv 37041  df-lfl 37080  df-lkr 37108  df-ldual 37146  df-oposet 37198  df-ol 37200  df-oml 37201  df-covers 37288  df-ats 37289  df-atl 37320  df-cvlat 37344  df-hlat 37373  df-llines 37520  df-lplanes 37521  df-lvols 37522  df-lines 37523  df-psubsp 37525  df-pmap 37526  df-padd 37818  df-lhyp 38010  df-laut 38011  df-ldil 38126  df-ltrn 38127  df-trl 38181  df-tgrp 38765  df-tendo 38777  df-edring 38779  df-dveca 39025  df-disoa 39051  df-dvech 39101  df-dib 39161  df-dic 39195  df-dih 39251  df-doch 39370  df-djh 39417  df-lcdual 39609  df-mapd 39647  df-hvmap 39779  df-hdmap1 39815  df-hdmap 39816
This theorem is referenced by:  hdmap14lem7  39896
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