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Theorem lsslindf 21376
Description: Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
lsslindf.u π‘ˆ = (LSubSpβ€˜π‘Š)
lsslindf.x 𝑋 = (π‘Š β†Ύs 𝑆)
Assertion
Ref Expression
lsslindf ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF π‘Š))
Proof of Theorem lsslindf
Dummy variables π‘˜ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rellindf 21354 . . . 4 Rel LIndF
21brrelex1i 5730 . . 3 (𝐹 LIndF 𝑋 β†’ 𝐹 ∈ V)
32a1i 11 . 2 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ (𝐹 LIndF 𝑋 β†’ 𝐹 ∈ V))
41brrelex1i 5730 . . 3 (𝐹 LIndF π‘Š β†’ 𝐹 ∈ V)
54a1i 11 . 2 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ (𝐹 LIndF π‘Š β†’ 𝐹 ∈ V))
6 simpr 485 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹:dom 𝐹⟢(Baseβ€˜π‘‹)) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘‹))
7 lsslindf.x . . . . . . . . 9 𝑋 = (π‘Š β†Ύs 𝑆)
8 eqid 2732 . . . . . . . . 9 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
97, 8ressbasss 17179 . . . . . . . 8 (Baseβ€˜π‘‹) βŠ† (Baseβ€˜π‘Š)
10 fss 6731 . . . . . . . 8 ((𝐹:dom 𝐹⟢(Baseβ€˜π‘‹) ∧ (Baseβ€˜π‘‹) βŠ† (Baseβ€˜π‘Š)) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
116, 9, 10sylancl 586 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹:dom 𝐹⟢(Baseβ€˜π‘‹)) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
12 ffn 6714 . . . . . . . . 9 (𝐹:dom 𝐹⟢(Baseβ€˜π‘Š) β†’ 𝐹 Fn dom 𝐹)
1312adantl 482 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š)) β†’ 𝐹 Fn dom 𝐹)
14 simp3 1138 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ ran 𝐹 βŠ† 𝑆)
15 lsslindf.u . . . . . . . . . . . . 13 π‘ˆ = (LSubSpβ€˜π‘Š)
168, 15lssss 20539 . . . . . . . . . . . 12 (𝑆 ∈ π‘ˆ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
17163ad2ant2 1134 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
187, 8ressbas2 17178 . . . . . . . . . . 11 (𝑆 βŠ† (Baseβ€˜π‘Š) β†’ 𝑆 = (Baseβ€˜π‘‹))
1917, 18syl 17 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ 𝑆 = (Baseβ€˜π‘‹))
2014, 19sseqtrd 4021 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ ran 𝐹 βŠ† (Baseβ€˜π‘‹))
2120adantr 481 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š)) β†’ ran 𝐹 βŠ† (Baseβ€˜π‘‹))
22 df-f 6544 . . . . . . . 8 (𝐹:dom 𝐹⟢(Baseβ€˜π‘‹) ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 βŠ† (Baseβ€˜π‘‹)))
2313, 21, 22sylanbrc 583 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š)) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘‹))
2411, 23impbida 799 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ (𝐹:dom 𝐹⟢(Baseβ€˜π‘‹) ↔ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š)))
2524adantr 481 . . . . 5 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (𝐹:dom 𝐹⟢(Baseβ€˜π‘‹) ↔ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š)))
26 simpl2 1192 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ 𝑆 ∈ π‘ˆ)
27 eqid 2732 . . . . . . . . . . . 12 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
287, 27resssca 17284 . . . . . . . . . . 11 (𝑆 ∈ π‘ˆ β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘‹))
2928eqcomd 2738 . . . . . . . . . 10 (𝑆 ∈ π‘ˆ β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘Š))
3026, 29syl 17 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘Š))
3130fveq2d 6892 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
3230fveq2d 6892 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
3332sneqd 4639 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ {(0gβ€˜(Scalarβ€˜π‘‹))} = {(0gβ€˜(Scalarβ€˜π‘Š))})
3431, 33difeq12d 4122 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ ((Baseβ€˜(Scalarβ€˜π‘‹)) βˆ– {(0gβ€˜(Scalarβ€˜π‘‹))}) = ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))
35 eqid 2732 . . . . . . . . . . . . 13 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
367, 35ressvsca 17285 . . . . . . . . . . . 12 (𝑆 ∈ π‘ˆ β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘‹))
3736eqcomd 2738 . . . . . . . . . . 11 (𝑆 ∈ π‘ˆ β†’ ( ·𝑠 β€˜π‘‹) = ( ·𝑠 β€˜π‘Š))
3826, 37syl 17 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ ( ·𝑠 β€˜π‘‹) = ( ·𝑠 β€˜π‘Š))
3938oveqd 7422 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (π‘˜( ·𝑠 β€˜π‘‹)(πΉβ€˜π‘₯)) = (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)))
40 simpl1 1191 . . . . . . . . . . 11 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ π‘Š ∈ LMod)
41 imassrn 6068 . . . . . . . . . . . 12 (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})) βŠ† ran 𝐹
42 simpl3 1193 . . . . . . . . . . . 12 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ ran 𝐹 βŠ† 𝑆)
4341, 42sstrid 3992 . . . . . . . . . . 11 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})) βŠ† 𝑆)
44 eqid 2732 . . . . . . . . . . . 12 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
45 eqid 2732 . . . . . . . . . . . 12 (LSpanβ€˜π‘‹) = (LSpanβ€˜π‘‹)
467, 44, 45, 15lsslsp 20618 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})) βŠ† 𝑆) β†’ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))) = ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))
4740, 26, 43, 46syl3anc 1371 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))) = ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))
4847eqcomd 2738 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))) = ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))
4939, 48eleq12d 2827 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ ((π‘˜( ·𝑠 β€˜π‘‹)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))) ↔ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))))
5049notbid 317 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (Β¬ (π‘˜( ·𝑠 β€˜π‘‹)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))) ↔ Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))))
5134, 50raleqbidv 3342 . . . . . 6 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘‹)) βˆ– {(0gβ€˜(Scalarβ€˜π‘‹))}) Β¬ (π‘˜( ·𝑠 β€˜π‘‹)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))) ↔ βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))))
5251ralbidv 3177 . . . . 5 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘‹)) βˆ– {(0gβ€˜(Scalarβ€˜π‘‹))}) Β¬ (π‘˜( ·𝑠 β€˜π‘‹)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))) ↔ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))))
5325, 52anbi12d 631 . . . 4 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ ((𝐹:dom 𝐹⟢(Baseβ€˜π‘‹) ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘‹)) βˆ– {(0gβ€˜(Scalarβ€˜π‘‹))}) Β¬ (π‘˜( ·𝑠 β€˜π‘‹)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})))) ↔ (𝐹:dom 𝐹⟢(Baseβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
547ovexi 7439 . . . . . 6 𝑋 ∈ V
5554a1i 11 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ 𝑋 ∈ V)
56 eqid 2732 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
57 eqid 2732 . . . . . 6 ( ·𝑠 β€˜π‘‹) = ( ·𝑠 β€˜π‘‹)
58 eqid 2732 . . . . . 6 (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘‹)
59 eqid 2732 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘‹))
60 eqid 2732 . . . . . 6 (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘‹))
6156, 57, 45, 58, 59, 60islindf 21358 . . . . 5 ((𝑋 ∈ V ∧ 𝐹 ∈ V) β†’ (𝐹 LIndF 𝑋 ↔ (𝐹:dom 𝐹⟢(Baseβ€˜π‘‹) ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘‹)) βˆ– {(0gβ€˜(Scalarβ€˜π‘‹))}) Β¬ (π‘˜( ·𝑠 β€˜π‘‹)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
6255, 61sylan 580 . . . 4 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (𝐹 LIndF 𝑋 ↔ (𝐹:dom 𝐹⟢(Baseβ€˜π‘‹) ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘‹)) βˆ– {(0gβ€˜(Scalarβ€˜π‘‹))}) Β¬ (π‘˜( ·𝑠 β€˜π‘‹)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
63 eqid 2732 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
64 eqid 2732 . . . . . 6 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
658, 35, 44, 27, 63, 64islindf 21358 . . . . 5 ((π‘Š ∈ LMod ∧ 𝐹 ∈ V) β†’ (𝐹 LIndF π‘Š ↔ (𝐹:dom 𝐹⟢(Baseβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
66653ad2antl1 1185 . . . 4 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (𝐹 LIndF π‘Š ↔ (𝐹:dom 𝐹⟢(Baseβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))))
6753, 62, 663bitr4d 310 . . 3 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF π‘Š))
6867ex 413 . 2 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ (𝐹 ∈ V β†’ (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF π‘Š)))
693, 5, 68pm5.21ndd 380 1 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF π‘Š))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆ– cdif 3944   βŠ† wss 3947  {csn 4627   class class class wbr 5147  dom cdm 5675  ran crn 5676   β€œ cima 5678   Fn wfn 6535  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140   β†Ύs cress 17169  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  LModclmod 20463  LSubSpclss 20534  LSpanclspn 20574   LIndF clindf 21350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-sca 17209  df-vsca 17210  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820  df-subg 18997  df-mgp 19982  df-ur 19999  df-ring 20051  df-lmod 20465  df-lss 20535  df-lsp 20575  df-lindf 21352
This theorem is referenced by:  lsslinds  21377
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