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Theorem lindfpropd 32773
Description: Property deduction for linearly independent families. (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)
lindfpropd.x (πœ‘ β†’ 𝑋 ∈ 𝐴)
Assertion
Ref Expression
lindfpropd (πœ‘ β†’ (𝑋 LIndF 𝐾 ↔ 𝑋 LIndF 𝐿))
Distinct variable groups:   π‘₯,𝐾,𝑦   π‘₯,𝐿,𝑦   π‘₯,𝑋,𝑦   πœ‘,π‘₯,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝑉(π‘₯,𝑦)   π‘Š(π‘₯,𝑦)

Proof of Theorem lindfpropd
Dummy variables 𝑖 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lindfpropd.2 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜(Scalarβ€˜πΎ)) = (Baseβ€˜(Scalarβ€˜πΏ)))
2 lindfpropd.3 . . . . . . . . 9 (πœ‘ β†’ (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΏ)))
32sneqd 4640 . . . . . . . 8 (πœ‘ β†’ {(0gβ€˜(Scalarβ€˜πΎ))} = {(0gβ€˜(Scalarβ€˜πΏ))})
41, 3difeq12d 4123 . . . . . . 7 (πœ‘ β†’ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))}) = ((Baseβ€˜(Scalarβ€˜πΏ)) βˆ– {(0gβ€˜(Scalarβ€˜πΏ))}))
54ad2antrr 723 . . . . . 6 (((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) β†’ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))}) = ((Baseβ€˜(Scalarβ€˜πΏ)) βˆ– {(0gβ€˜(Scalarβ€˜πΏ))}))
6 simplll 772 . . . . . . . . 9 ((((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))})) β†’ πœ‘)
7 simpr 484 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))})) β†’ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))}))
87eldifad 3960 . . . . . . . . 9 ((((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))})) β†’ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜πΎ)))
9 simpr 484 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) β†’ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ))
109ffvelcdmda 7086 . . . . . . . . . 10 (((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) β†’ (π‘‹β€˜π‘–) ∈ (Baseβ€˜πΎ))
1110adantr 480 . . . . . . . . 9 ((((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))})) β†’ (π‘‹β€˜π‘–) ∈ (Baseβ€˜πΎ))
12 lindfpropd.6 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
1312oveqrspc2v 7439 . . . . . . . . 9 ((πœ‘ ∧ (π‘˜ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ (π‘‹β€˜π‘–) ∈ (Baseβ€˜πΎ))) β†’ (π‘˜( ·𝑠 β€˜πΎ)(π‘‹β€˜π‘–)) = (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)))
146, 8, 11, 13syl12anc 834 . . . . . . . 8 ((((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))})) β†’ (π‘˜( ·𝑠 β€˜πΎ)(π‘‹β€˜π‘–)) = (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)))
15 eqidd 2732 . . . . . . . . . . 11 (πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΎ))
16 lindfpropd.1 . . . . . . . . . . 11 (πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))
17 ssidd 4005 . . . . . . . . . . 11 (πœ‘ β†’ (Baseβ€˜πΎ) βŠ† (Baseβ€˜πΎ))
18 lindfpropd.4 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
19 lindfpropd.5 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ (Baseβ€˜πΎ))
20 eqidd 2732 . . . . . . . . . . 11 (πœ‘ β†’ (Baseβ€˜(Scalarβ€˜πΎ)) = (Baseβ€˜(Scalarβ€˜πΎ)))
21 lindfpropd.k . . . . . . . . . . 11 (πœ‘ β†’ 𝐾 ∈ 𝑉)
22 lindfpropd.l . . . . . . . . . . 11 (πœ‘ β†’ 𝐿 ∈ π‘Š)
2315, 16, 17, 18, 19, 12, 20, 1, 21, 22lsppropd 20774 . . . . . . . . . 10 (πœ‘ β†’ (LSpanβ€˜πΎ) = (LSpanβ€˜πΏ))
2423fveq1d 6893 . . . . . . . . 9 (πœ‘ β†’ ((LSpanβ€˜πΎ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))) = ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))))
2524ad3antrrr 727 . . . . . . . 8 ((((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))})) β†’ ((LSpanβ€˜πΎ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))) = ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))))
2614, 25eleq12d 2826 . . . . . . 7 ((((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))})) β†’ ((π‘˜( ·𝑠 β€˜πΎ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΎ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))) ↔ (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖})))))
2726notbid 318 . . . . . 6 ((((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))})) β†’ (Β¬ (π‘˜( ·𝑠 β€˜πΎ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΎ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))) ↔ Β¬ (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖})))))
285, 27raleqbidva 3326 . . . . 5 (((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) ∧ 𝑖 ∈ dom 𝑋) β†’ (βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))}) Β¬ (π‘˜( ·𝑠 β€˜πΎ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΎ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))) ↔ βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΏ)) βˆ– {(0gβ€˜(Scalarβ€˜πΏ))}) Β¬ (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖})))))
2928ralbidva 3174 . . . 4 ((πœ‘ ∧ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ)) β†’ (βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))}) Β¬ (π‘˜( ·𝑠 β€˜πΎ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΎ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))) ↔ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΏ)) βˆ– {(0gβ€˜(Scalarβ€˜πΏ))}) Β¬ (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖})))))
3029pm5.32da 578 . . 3 (πœ‘ β†’ ((𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ) ∧ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))}) Β¬ (π‘˜( ·𝑠 β€˜πΎ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΎ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖})))) ↔ (𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ) ∧ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΏ)) βˆ– {(0gβ€˜(Scalarβ€˜πΏ))}) Β¬ (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))))))
3116feq3d 6704 . . . 4 (πœ‘ β†’ (𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ) ↔ 𝑋:dom π‘‹βŸΆ(Baseβ€˜πΏ)))
3231anbi1d 629 . . 3 (πœ‘ β†’ ((𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ) ∧ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΏ)) βˆ– {(0gβ€˜(Scalarβ€˜πΏ))}) Β¬ (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖})))) ↔ (𝑋:dom π‘‹βŸΆ(Baseβ€˜πΏ) ∧ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΏ)) βˆ– {(0gβ€˜(Scalarβ€˜πΏ))}) Β¬ (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))))))
3330, 32bitrd 279 . 2 (πœ‘ β†’ ((𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ) ∧ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))}) Β¬ (π‘˜( ·𝑠 β€˜πΎ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΎ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖})))) ↔ (𝑋:dom π‘‹βŸΆ(Baseβ€˜πΏ) ∧ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΏ)) βˆ– {(0gβ€˜(Scalarβ€˜πΏ))}) Β¬ (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))))))
34 lindfpropd.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐴)
35 eqid 2731 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
36 eqid 2731 . . . 4 ( ·𝑠 β€˜πΎ) = ( ·𝑠 β€˜πΎ)
37 eqid 2731 . . . 4 (LSpanβ€˜πΎ) = (LSpanβ€˜πΎ)
38 eqid 2731 . . . 4 (Scalarβ€˜πΎ) = (Scalarβ€˜πΎ)
39 eqid 2731 . . . 4 (Baseβ€˜(Scalarβ€˜πΎ)) = (Baseβ€˜(Scalarβ€˜πΎ))
40 eqid 2731 . . . 4 (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΎ))
4135, 36, 37, 38, 39, 40islindf 21587 . . 3 ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) β†’ (𝑋 LIndF 𝐾 ↔ (𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ) ∧ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))}) Β¬ (π‘˜( ·𝑠 β€˜πΎ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΎ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))))))
4221, 34, 41syl2anc 583 . 2 (πœ‘ β†’ (𝑋 LIndF 𝐾 ↔ (𝑋:dom π‘‹βŸΆ(Baseβ€˜πΎ) ∧ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΎ)) βˆ– {(0gβ€˜(Scalarβ€˜πΎ))}) Β¬ (π‘˜( ·𝑠 β€˜πΎ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΎ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))))))
43 eqid 2731 . . . 4 (Baseβ€˜πΏ) = (Baseβ€˜πΏ)
44 eqid 2731 . . . 4 ( ·𝑠 β€˜πΏ) = ( ·𝑠 β€˜πΏ)
45 eqid 2731 . . . 4 (LSpanβ€˜πΏ) = (LSpanβ€˜πΏ)
46 eqid 2731 . . . 4 (Scalarβ€˜πΏ) = (Scalarβ€˜πΏ)
47 eqid 2731 . . . 4 (Baseβ€˜(Scalarβ€˜πΏ)) = (Baseβ€˜(Scalarβ€˜πΏ))
48 eqid 2731 . . . 4 (0gβ€˜(Scalarβ€˜πΏ)) = (0gβ€˜(Scalarβ€˜πΏ))
4943, 44, 45, 46, 47, 48islindf 21587 . . 3 ((𝐿 ∈ π‘Š ∧ 𝑋 ∈ 𝐴) β†’ (𝑋 LIndF 𝐿 ↔ (𝑋:dom π‘‹βŸΆ(Baseβ€˜πΏ) ∧ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΏ)) βˆ– {(0gβ€˜(Scalarβ€˜πΏ))}) Β¬ (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))))))
5022, 34, 49syl2anc 583 . 2 (πœ‘ β†’ (𝑋 LIndF 𝐿 ↔ (𝑋:dom π‘‹βŸΆ(Baseβ€˜πΏ) ∧ βˆ€π‘– ∈ dom π‘‹βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜πΏ)) βˆ– {(0gβ€˜(Scalarβ€˜πΏ))}) Β¬ (π‘˜( ·𝑠 β€˜πΏ)(π‘‹β€˜π‘–)) ∈ ((LSpanβ€˜πΏ)β€˜(𝑋 β€œ (dom 𝑋 βˆ– {𝑖}))))))
5133, 42, 503bitr4d 311 1 (πœ‘ β†’ (𝑋 LIndF 𝐾 ↔ 𝑋 LIndF 𝐿))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060   βˆ– cdif 3945  {csn 4628   class class class wbr 5148  dom cdm 5676   β€œ cima 5679  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  +gcplusg 17202  Scalarcsca 17205   ·𝑠 cvsca 17206  0gc0g 17390  LSpanclspn 20727   LIndF clindf 21579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-lss 20688  df-lsp 20728  df-lindf 21581
This theorem is referenced by:  lindspropd  32774
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