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Theorem lindspropd 32218
Description: Property deduction for linearly independent sets. (Contributed by Thierry Arnoux, 16-Jul-2023.)
Hypotheses
Ref Expression
lindfpropd.1 (πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))
lindfpropd.2 (πœ‘ β†’ (Baseβ€˜(Scalarβ€˜πΎ)) = (Baseβ€˜(Scalarβ€˜πΏ)))
lindfpropd.3 (πœ‘ β†’ (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΏ)))
lindfpropd.4 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
lindfpropd.5 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ (Baseβ€˜πΎ))
lindfpropd.6 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
lindfpropd.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
lindfpropd.l (πœ‘ β†’ 𝐿 ∈ π‘Š)
Assertion
Ref Expression
lindspropd (πœ‘ β†’ (LIndSβ€˜πΎ) = (LIndSβ€˜πΏ))
Distinct variable groups:   π‘₯,𝐾,𝑦   π‘₯,𝐿,𝑦   πœ‘,π‘₯,𝑦
Allowed substitution hints:   𝑉(π‘₯,𝑦)   π‘Š(π‘₯,𝑦)

Proof of Theorem lindspropd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 lindfpropd.1 . . . . 5 (πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))
21sseq2d 3977 . . . 4 (πœ‘ β†’ (𝑧 βŠ† (Baseβ€˜πΎ) ↔ 𝑧 βŠ† (Baseβ€˜πΏ)))
3 lindfpropd.2 . . . . 5 (πœ‘ β†’ (Baseβ€˜(Scalarβ€˜πΎ)) = (Baseβ€˜(Scalarβ€˜πΏ)))
4 lindfpropd.3 . . . . 5 (πœ‘ β†’ (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΏ)))
5 lindfpropd.4 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
6 lindfpropd.5 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ (Baseβ€˜πΎ))
7 lindfpropd.6 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
8 lindfpropd.k . . . . 5 (πœ‘ β†’ 𝐾 ∈ 𝑉)
9 lindfpropd.l . . . . 5 (πœ‘ β†’ 𝐿 ∈ π‘Š)
10 vex 3448 . . . . . . 7 𝑧 ∈ V
1110a1i 11 . . . . . 6 (πœ‘ β†’ 𝑧 ∈ V)
1211resiexd 7167 . . . . 5 (πœ‘ β†’ ( I β†Ύ 𝑧) ∈ V)
131, 3, 4, 5, 6, 7, 8, 9, 12lindfpropd 32217 . . . 4 (πœ‘ β†’ (( I β†Ύ 𝑧) LIndF 𝐾 ↔ ( I β†Ύ 𝑧) LIndF 𝐿))
142, 13anbi12d 632 . . 3 (πœ‘ β†’ ((𝑧 βŠ† (Baseβ€˜πΎ) ∧ ( I β†Ύ 𝑧) LIndF 𝐾) ↔ (𝑧 βŠ† (Baseβ€˜πΏ) ∧ ( I β†Ύ 𝑧) LIndF 𝐿)))
15 eqid 2733 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1615islinds 21231 . . . 4 (𝐾 ∈ 𝑉 β†’ (𝑧 ∈ (LIndSβ€˜πΎ) ↔ (𝑧 βŠ† (Baseβ€˜πΎ) ∧ ( I β†Ύ 𝑧) LIndF 𝐾)))
178, 16syl 17 . . 3 (πœ‘ β†’ (𝑧 ∈ (LIndSβ€˜πΎ) ↔ (𝑧 βŠ† (Baseβ€˜πΎ) ∧ ( I β†Ύ 𝑧) LIndF 𝐾)))
18 eqid 2733 . . . . 5 (Baseβ€˜πΏ) = (Baseβ€˜πΏ)
1918islinds 21231 . . . 4 (𝐿 ∈ π‘Š β†’ (𝑧 ∈ (LIndSβ€˜πΏ) ↔ (𝑧 βŠ† (Baseβ€˜πΏ) ∧ ( I β†Ύ 𝑧) LIndF 𝐿)))
209, 19syl 17 . . 3 (πœ‘ β†’ (𝑧 ∈ (LIndSβ€˜πΏ) ↔ (𝑧 βŠ† (Baseβ€˜πΏ) ∧ ( I β†Ύ 𝑧) LIndF 𝐿)))
2114, 17, 203bitr4d 311 . 2 (πœ‘ β†’ (𝑧 ∈ (LIndSβ€˜πΎ) ↔ 𝑧 ∈ (LIndSβ€˜πΏ)))
2221eqrdv 2731 1 (πœ‘ β†’ (LIndSβ€˜πΎ) = (LIndSβ€˜πΏ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3444   βŠ† wss 3911   class class class wbr 5106   I cid 5531   β†Ύ cres 5636  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Scalarcsca 17141   ·𝑠 cvsca 17142  0gc0g 17326   LIndF clindf 21226  LIndSclinds 21227
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-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-lss 20408  df-lsp 20448  df-lindf 21228  df-linds 21229
This theorem is referenced by:  fedgmullem2  32382
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