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Theorem islno 30006
Description: The predicate "is a linear operator." (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnoval.1 𝑋 = (BaseSetβ€˜π‘ˆ)
lnoval.2 π‘Œ = (BaseSetβ€˜π‘Š)
lnoval.3 𝐺 = ( +𝑣 β€˜π‘ˆ)
lnoval.4 𝐻 = ( +𝑣 β€˜π‘Š)
lnoval.5 𝑅 = ( ·𝑠OLD β€˜π‘ˆ)
lnoval.6 𝑆 = ( ·𝑠OLD β€˜π‘Š)
lnoval.7 𝐿 = (π‘ˆ LnOp π‘Š)
Assertion
Ref Expression
islno ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ (𝑇 ∈ 𝐿 ↔ (𝑇:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‡β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‡β€˜π‘¦))𝐻(π‘‡β€˜π‘§)))))
Distinct variable groups:   π‘₯,𝑦,𝑧,π‘ˆ   π‘₯,π‘Š,𝑦,𝑧   𝑦,𝑋,𝑧   π‘₯,𝑇,𝑦,𝑧
Allowed substitution hints:   𝑅(π‘₯,𝑦,𝑧)   𝑆(π‘₯,𝑦,𝑧)   𝐺(π‘₯,𝑦,𝑧)   𝐻(π‘₯,𝑦,𝑧)   𝐿(π‘₯,𝑦,𝑧)   𝑋(π‘₯)   π‘Œ(π‘₯,𝑦,𝑧)

Proof of Theorem islno
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 lnoval.1 . . . 4 𝑋 = (BaseSetβ€˜π‘ˆ)
2 lnoval.2 . . . 4 π‘Œ = (BaseSetβ€˜π‘Š)
3 lnoval.3 . . . 4 𝐺 = ( +𝑣 β€˜π‘ˆ)
4 lnoval.4 . . . 4 𝐻 = ( +𝑣 β€˜π‘Š)
5 lnoval.5 . . . 4 𝑅 = ( ·𝑠OLD β€˜π‘ˆ)
6 lnoval.6 . . . 4 𝑆 = ( ·𝑠OLD β€˜π‘Š)
7 lnoval.7 . . . 4 𝐿 = (π‘ˆ LnOp π‘Š)
81, 2, 3, 4, 5, 6, 7lnoval 30005 . . 3 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝐿 = {𝑀 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘€β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘€β€˜π‘¦))𝐻(π‘€β€˜π‘§))})
98eleq2d 2820 . 2 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ (𝑇 ∈ 𝐿 ↔ 𝑇 ∈ {𝑀 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘€β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘€β€˜π‘¦))𝐻(π‘€β€˜π‘§))}))
10 fveq1 6891 . . . . . . 7 (𝑀 = 𝑇 β†’ (π‘€β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = (π‘‡β€˜((π‘₯𝑅𝑦)𝐺𝑧)))
11 fveq1 6891 . . . . . . . . 9 (𝑀 = 𝑇 β†’ (π‘€β€˜π‘¦) = (π‘‡β€˜π‘¦))
1211oveq2d 7425 . . . . . . . 8 (𝑀 = 𝑇 β†’ (π‘₯𝑆(π‘€β€˜π‘¦)) = (π‘₯𝑆(π‘‡β€˜π‘¦)))
13 fveq1 6891 . . . . . . . 8 (𝑀 = 𝑇 β†’ (π‘€β€˜π‘§) = (π‘‡β€˜π‘§))
1412, 13oveq12d 7427 . . . . . . 7 (𝑀 = 𝑇 β†’ ((π‘₯𝑆(π‘€β€˜π‘¦))𝐻(π‘€β€˜π‘§)) = ((π‘₯𝑆(π‘‡β€˜π‘¦))𝐻(π‘‡β€˜π‘§)))
1510, 14eqeq12d 2749 . . . . . 6 (𝑀 = 𝑇 β†’ ((π‘€β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘€β€˜π‘¦))𝐻(π‘€β€˜π‘§)) ↔ (π‘‡β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‡β€˜π‘¦))𝐻(π‘‡β€˜π‘§))))
16152ralbidv 3219 . . . . 5 (𝑀 = 𝑇 β†’ (βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘€β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘€β€˜π‘¦))𝐻(π‘€β€˜π‘§)) ↔ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‡β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‡β€˜π‘¦))𝐻(π‘‡β€˜π‘§))))
1716ralbidv 3178 . . . 4 (𝑀 = 𝑇 β†’ (βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘€β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘€β€˜π‘¦))𝐻(π‘€β€˜π‘§)) ↔ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‡β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‡β€˜π‘¦))𝐻(π‘‡β€˜π‘§))))
1817elrab 3684 . . 3 (𝑇 ∈ {𝑀 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘€β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘€β€˜π‘¦))𝐻(π‘€β€˜π‘§))} ↔ (𝑇 ∈ (π‘Œ ↑m 𝑋) ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‡β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‡β€˜π‘¦))𝐻(π‘‡β€˜π‘§))))
192fvexi 6906 . . . . 5 π‘Œ ∈ V
201fvexi 6906 . . . . 5 𝑋 ∈ V
2119, 20elmap 8865 . . . 4 (𝑇 ∈ (π‘Œ ↑m 𝑋) ↔ 𝑇:π‘‹βŸΆπ‘Œ)
2221anbi1i 625 . . 3 ((𝑇 ∈ (π‘Œ ↑m 𝑋) ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‡β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‡β€˜π‘¦))𝐻(π‘‡β€˜π‘§))) ↔ (𝑇:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‡β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‡β€˜π‘¦))𝐻(π‘‡β€˜π‘§))))
2318, 22bitri 275 . 2 (𝑇 ∈ {𝑀 ∈ (π‘Œ ↑m 𝑋) ∣ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘€β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘€β€˜π‘¦))𝐻(π‘€β€˜π‘§))} ↔ (𝑇:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‡β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‡β€˜π‘¦))𝐻(π‘‡β€˜π‘§))))
249, 23bitrdi 287 1 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ (𝑇 ∈ 𝐿 ↔ (𝑇:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 (π‘‡β€˜((π‘₯𝑅𝑦)𝐺𝑧)) = ((π‘₯𝑆(π‘‡β€˜π‘¦))𝐻(π‘‡β€˜π‘§)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  β„‚cc 11108  NrmCVeccnv 29837   +𝑣 cpv 29838  BaseSetcba 29839   ·𝑠OLD cns 29840   LnOp clno 29993
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-lno 29997
This theorem is referenced by:  lnolin  30007  lnof  30008  lnocoi  30010  0lno  30043  ipblnfi  30108
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