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Theorem lsslindf 21725
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 21703 . . . 4 Rel LIndF
21brrelex1i 5725 . . 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 5725 . . 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 484 . . . . . . . 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 2726 . . . . . . . . 9 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
97, 8ressbasss 17192 . . . . . . . 8 (Baseβ€˜π‘‹) βŠ† (Baseβ€˜π‘Š)
10 fss 6728 . . . . . . . 8 ((𝐹:dom 𝐹⟢(Baseβ€˜π‘‹) ∧ (Baseβ€˜π‘‹) βŠ† (Baseβ€˜π‘Š)) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
116, 9, 10sylancl 585 . . . . . . 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 6711 . . . . . . . . 9 (𝐹:dom 𝐹⟢(Baseβ€˜π‘Š) β†’ 𝐹 Fn dom 𝐹)
1312adantl 481 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š)) β†’ 𝐹 Fn dom 𝐹)
14 simp3 1135 . . . . . . . . . 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 20783 . . . . . . . . . . . 12 (𝑆 ∈ π‘ˆ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
17163ad2ant2 1131 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
187, 8ressbas2 17191 . . . . . . . . . . 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 4017 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ ran 𝐹 βŠ† (Baseβ€˜π‘‹))
2120adantr 480 . . . . . . . 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 6541 . . . . . . . 8 (𝐹:dom 𝐹⟢(Baseβ€˜π‘‹) ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 βŠ† (Baseβ€˜π‘‹)))
2313, 21, 22sylanbrc 582 . . . . . . 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 798 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ (𝐹:dom 𝐹⟢(Baseβ€˜π‘‹) ↔ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š)))
2524adantr 480 . . . . 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 1189 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ 𝑆 ∈ π‘ˆ)
27 eqid 2726 . . . . . . . . . . . 12 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
287, 27resssca 17297 . . . . . . . . . . 11 (𝑆 ∈ π‘ˆ β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘‹))
2928eqcomd 2732 . . . . . . . . . 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 6889 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
3230fveq2d 6889 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
3332sneqd 4635 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ {(0gβ€˜(Scalarβ€˜π‘‹))} = {(0gβ€˜(Scalarβ€˜π‘Š))})
3431, 33difeq12d 4118 . . . . . . 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 2726 . . . . . . . . . . . . 13 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
367, 35ressvsca 17298 . . . . . . . . . . . 12 (𝑆 ∈ π‘ˆ β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘‹))
3736eqcomd 2732 . . . . . . . . . . 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 1188 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ π‘Š ∈ LMod)
41 imassrn 6064 . . . . . . . . . . 11 (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})) βŠ† ran 𝐹
42 simpl3 1190 . . . . . . . . . . 11 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ ran 𝐹 βŠ† 𝑆)
4341, 42sstrid 3988 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})) βŠ† 𝑆)
44 eqid 2726 . . . . . . . . . . 11 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
45 eqid 2726 . . . . . . . . . . 11 (LSpanβ€˜π‘‹) = (LSpanβ€˜π‘‹)
467, 44, 45, 15lsslsp 20862 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ (𝐹 β€œ (dom 𝐹 βˆ– {π‘₯})) βŠ† 𝑆) β†’ ((LSpanβ€˜π‘‹)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))) = ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {π‘₯}))))
4740, 26, 43, 46syl3anc 1368 . . . . . . . . 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‘₯}))))
4839, 47eleq12d 2821 . . . . . . . 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‘₯})))))
4948notbid 318 . . . . . . 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‘₯})))))
5034, 49raleqbidv 3336 . . . . . 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‘₯})))))
5150ralbidv 3171 . . . . 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‘₯})))))
5225, 51anbi12d 630 . . . 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‘₯}))))))
537ovexi 7439 . . . . . 6 𝑋 ∈ V
5453a1i 11 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ 𝑋 ∈ V)
55 eqid 2726 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
56 eqid 2726 . . . . . 6 ( ·𝑠 β€˜π‘‹) = ( ·𝑠 β€˜π‘‹)
57 eqid 2726 . . . . . 6 (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘‹)
58 eqid 2726 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘‹))
59 eqid 2726 . . . . . 6 (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘‹))
6055, 56, 45, 57, 58, 59islindf 21707 . . . . 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‘₯}))))))
6154, 60sylan 579 . . . 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‘₯}))))))
62 eqid 2726 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
63 eqid 2726 . . . . . 6 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
648, 35, 44, 27, 62, 63islindf 21707 . . . . 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‘₯}))))))
65643ad2antl1 1182 . . . 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‘₯}))))))
6652, 61, 653bitr4d 311 . . 3 (((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) ∧ 𝐹 ∈ V) β†’ (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF π‘Š))
6766ex 412 . 2 ((π‘Š ∈ LMod ∧ 𝑆 ∈ π‘ˆ ∧ ran 𝐹 βŠ† 𝑆) β†’ (𝐹 ∈ V β†’ (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF π‘Š)))
683, 5, 67pm5.21ndd 379 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   βˆ– cdif 3940   βŠ† wss 3943  {csn 4623   class class class wbr 5141  dom cdm 5669  ran crn 5670   β€œ cima 5672   Fn wfn 6532  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153   β†Ύs cress 17182  Scalarcsca 17209   ·𝑠 cvsca 17210  0gc0g 17394  LModclmod 20706  LSubSpclss 20778  LSpanclspn 20818   LIndF clindf 21699
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  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  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-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  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-pss 3962  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-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  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-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  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-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-sca 17222  df-vsca 17223  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19050  df-mgp 20040  df-ur 20087  df-ring 20140  df-lmod 20708  df-lss 20779  df-lsp 20819  df-lindf 21701
This theorem is referenced by:  lsslinds  21726
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