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Theorem lindspropd 33005
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 4009 . . . 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 3472 . . . . . . 7 𝑧 ∈ V
1110a1i 11 . . . . . 6 (πœ‘ β†’ 𝑧 ∈ V)
1211resiexd 7213 . . . . 5 (πœ‘ β†’ ( I β†Ύ 𝑧) ∈ V)
131, 3, 4, 5, 6, 7, 8, 9, 12lindfpropd 33004 . . . 4 (πœ‘ β†’ (( I β†Ύ 𝑧) LIndF 𝐾 ↔ ( I β†Ύ 𝑧) LIndF 𝐿))
142, 13anbi12d 630 . . 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 2726 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
1615islinds 21704 . . . 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 2726 . . . . 5 (Baseβ€˜πΏ) = (Baseβ€˜πΏ)
1918islinds 21704 . . . 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 2724 1 (πœ‘ β†’ (LIndSβ€˜πΎ) = (LIndSβ€˜πΏ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   βŠ† wss 3943   class class class wbr 5141   I cid 5566   β†Ύ cres 5671  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  Scalarcsca 17209   ·𝑠 cvsca 17210  0gc0g 17394   LIndF clindf 21699  LIndSclinds 21700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-lss 20779  df-lsp 20819  df-lindf 21701  df-linds 21702
This theorem is referenced by:  fedgmullem2  33233
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