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Theorem lspsneq 20880
Description: Equal spans of singletons must have proportional vectors. See lspsnss2 20760 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
Hypotheses
Ref Expression
lspsneq.v 𝑉 = (Baseβ€˜π‘Š)
lspsneq.s 𝑆 = (Scalarβ€˜π‘Š)
lspsneq.k 𝐾 = (Baseβ€˜π‘†)
lspsneq.o 0 = (0gβ€˜π‘†)
lspsneq.t Β· = ( ·𝑠 β€˜π‘Š)
lspsneq.n 𝑁 = (LSpanβ€˜π‘Š)
lspsneq.w (πœ‘ β†’ π‘Š ∈ LVec)
lspsneq.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
lspsneq.y (πœ‘ β†’ π‘Œ ∈ 𝑉)
Assertion
Ref Expression
lspsneq (πœ‘ β†’ ((π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ (𝐾 βˆ– { 0 })𝑋 = (π‘˜ Β· π‘Œ)))
Distinct variable groups:   π‘˜,𝐾   0 ,π‘˜   Β· ,π‘˜   π‘˜,𝑋   π‘˜,π‘Œ
Allowed substitution hints:   πœ‘(π‘˜)   𝑆(π‘˜)   𝑁(π‘˜)   𝑉(π‘˜)   π‘Š(π‘˜)
Proof of Theorem lspsneq
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 lspsneq.w . . . . . . . . . 10 (πœ‘ β†’ π‘Š ∈ LVec)
2 lveclmod 20861 . . . . . . . . . 10 (π‘Š ∈ LVec β†’ π‘Š ∈ LMod)
31, 2syl 17 . . . . . . . . 9 (πœ‘ β†’ π‘Š ∈ LMod)
4 lspsneq.s . . . . . . . . . 10 𝑆 = (Scalarβ€˜π‘Š)
54lmodring 20622 . . . . . . . . 9 (π‘Š ∈ LMod β†’ 𝑆 ∈ Ring)
6 lspsneq.k . . . . . . . . . 10 𝐾 = (Baseβ€˜π‘†)
7 eqid 2732 . . . . . . . . . 10 (1rβ€˜π‘†) = (1rβ€˜π‘†)
86, 7ringidcl 20154 . . . . . . . . 9 (𝑆 ∈ Ring β†’ (1rβ€˜π‘†) ∈ 𝐾)
93, 5, 83syl 18 . . . . . . . 8 (πœ‘ β†’ (1rβ€˜π‘†) ∈ 𝐾)
104lvecdrng 20860 . . . . . . . . 9 (π‘Š ∈ LVec β†’ 𝑆 ∈ DivRing)
11 lspsneq.o . . . . . . . . . 10 0 = (0gβ€˜π‘†)
1211, 7drngunz 20519 . . . . . . . . 9 (𝑆 ∈ DivRing β†’ (1rβ€˜π‘†) β‰  0 )
131, 10, 123syl 18 . . . . . . . 8 (πœ‘ β†’ (1rβ€˜π‘†) β‰  0 )
14 eldifsn 4790 . . . . . . . 8 ((1rβ€˜π‘†) ∈ (𝐾 βˆ– { 0 }) ↔ ((1rβ€˜π‘†) ∈ 𝐾 ∧ (1rβ€˜π‘†) β‰  0 ))
159, 13, 14sylanbrc 583 . . . . . . 7 (πœ‘ β†’ (1rβ€˜π‘†) ∈ (𝐾 βˆ– { 0 }))
1615ad2antrr 724 . . . . . 6 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ = (0gβ€˜π‘Š)) β†’ (1rβ€˜π‘†) ∈ (𝐾 βˆ– { 0 }))
17 lspsneq.v . . . . . . . . . 10 𝑉 = (Baseβ€˜π‘Š)
18 eqid 2732 . . . . . . . . . 10 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
1917, 18lmod0vcl 20645 . . . . . . . . 9 (π‘Š ∈ LMod β†’ (0gβ€˜π‘Š) ∈ 𝑉)
20 lspsneq.t . . . . . . . . . 10 Β· = ( ·𝑠 β€˜π‘Š)
2117, 4, 20, 7lmodvs1 20644 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ((1rβ€˜π‘†) Β· (0gβ€˜π‘Š)) = (0gβ€˜π‘Š))
223, 19, 21syl2anc2 585 . . . . . . . 8 (πœ‘ β†’ ((1rβ€˜π‘†) Β· (0gβ€˜π‘Š)) = (0gβ€˜π‘Š))
2322ad2antrr 724 . . . . . . 7 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ = (0gβ€˜π‘Š)) β†’ ((1rβ€˜π‘†) Β· (0gβ€˜π‘Š)) = (0gβ€˜π‘Š))
24 oveq2 7419 . . . . . . . 8 (π‘Œ = (0gβ€˜π‘Š) β†’ ((1rβ€˜π‘†) Β· π‘Œ) = ((1rβ€˜π‘†) Β· (0gβ€˜π‘Š)))
2524adantl 482 . . . . . . 7 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ = (0gβ€˜π‘Š)) β†’ ((1rβ€˜π‘†) Β· π‘Œ) = ((1rβ€˜π‘†) Β· (0gβ€˜π‘Š)))
26 lspsneq.n . . . . . . . . 9 𝑁 = (LSpanβ€˜π‘Š)
273adantr 481 . . . . . . . . 9 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ π‘Š ∈ LMod)
28 lspsneq.x . . . . . . . . . 10 (πœ‘ β†’ 𝑋 ∈ 𝑉)
2928adantr 481 . . . . . . . . 9 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ 𝑋 ∈ 𝑉)
30 lspsneq.y . . . . . . . . . 10 (πœ‘ β†’ π‘Œ ∈ 𝑉)
3130adantr 481 . . . . . . . . 9 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ π‘Œ ∈ 𝑉)
32 simpr 485 . . . . . . . . 9 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}))
3317, 18, 26, 27, 29, 31, 32lspsneq0b 20768 . . . . . . . 8 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (𝑋 = (0gβ€˜π‘Š) ↔ π‘Œ = (0gβ€˜π‘Š)))
3433biimpar 478 . . . . . . 7 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ = (0gβ€˜π‘Š)) β†’ 𝑋 = (0gβ€˜π‘Š))
3523, 25, 343eqtr4rd 2783 . . . . . 6 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ = (0gβ€˜π‘Š)) β†’ 𝑋 = ((1rβ€˜π‘†) Β· π‘Œ))
36 oveq1 7418 . . . . . . 7 (𝑗 = (1rβ€˜π‘†) β†’ (𝑗 Β· π‘Œ) = ((1rβ€˜π‘†) Β· π‘Œ))
3736rspceeqv 3633 . . . . . 6 (((1rβ€˜π‘†) ∈ (𝐾 βˆ– { 0 }) ∧ 𝑋 = ((1rβ€˜π‘†) Β· π‘Œ)) β†’ βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ))
3816, 35, 37syl2anc 584 . . . . 5 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ = (0gβ€˜π‘Š)) β†’ βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ))
39 eqimss 4040 . . . . . . . . . 10 ((π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}) β†’ (π‘β€˜{𝑋}) βŠ† (π‘β€˜{π‘Œ}))
4039adantl 482 . . . . . . . . 9 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (π‘β€˜{𝑋}) βŠ† (π‘β€˜{π‘Œ}))
41 eqid 2732 . . . . . . . . . 10 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
4217, 41, 26lspsncl 20732 . . . . . . . . . . . 12 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
433, 30, 42syl2anc 584 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
4443adantr 481 . . . . . . . . . 10 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
4517, 41, 26, 27, 44, 29lspsnel5 20750 . . . . . . . . 9 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (𝑋 ∈ (π‘β€˜{π‘Œ}) ↔ (π‘β€˜{𝑋}) βŠ† (π‘β€˜{π‘Œ})))
4640, 45mpbird 256 . . . . . . . 8 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ 𝑋 ∈ (π‘β€˜{π‘Œ}))
474, 6, 17, 20, 26lspsnel 20758 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 ∈ (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘— ∈ 𝐾 𝑋 = (𝑗 Β· π‘Œ)))
4827, 31, 47syl2anc 584 . . . . . . . 8 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (𝑋 ∈ (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘— ∈ 𝐾 𝑋 = (𝑗 Β· π‘Œ)))
4946, 48mpbid 231 . . . . . . 7 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ βˆƒπ‘— ∈ 𝐾 𝑋 = (𝑗 Β· π‘Œ))
5049adantr 481 . . . . . 6 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) β†’ βˆƒπ‘— ∈ 𝐾 𝑋 = (𝑗 Β· π‘Œ))
51 simprl 769 . . . . . . 7 ((((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) ∧ (𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ))) β†’ 𝑗 ∈ 𝐾)
52 simpr 485 . . . . . . . . . . 11 ((𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ)) β†’ 𝑋 = (𝑗 Β· π‘Œ))
5352adantl 482 . . . . . . . . . 10 ((((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) ∧ (𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ))) β†’ 𝑋 = (𝑗 Β· π‘Œ))
5433biimpd 228 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (𝑋 = (0gβ€˜π‘Š) β†’ π‘Œ = (0gβ€˜π‘Š)))
5554necon3d 2961 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (π‘Œ β‰  (0gβ€˜π‘Š) β†’ 𝑋 β‰  (0gβ€˜π‘Š)))
5655imp 407 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) β†’ 𝑋 β‰  (0gβ€˜π‘Š))
5756adantr 481 . . . . . . . . . 10 ((((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) ∧ (𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ))) β†’ 𝑋 β‰  (0gβ€˜π‘Š))
5853, 57eqnetrrd 3009 . . . . . . . . 9 ((((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) ∧ (𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ))) β†’ (𝑗 Β· π‘Œ) β‰  (0gβ€˜π‘Š))
591adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ π‘Š ∈ LVec)
6059ad2antrr 724 . . . . . . . . . 10 ((((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) ∧ (𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ))) β†’ π‘Š ∈ LVec)
6131ad2antrr 724 . . . . . . . . . 10 ((((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) ∧ (𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ))) β†’ π‘Œ ∈ 𝑉)
6217, 20, 4, 6, 11, 18, 60, 51, 61lvecvsn0 20867 . . . . . . . . 9 ((((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) ∧ (𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ))) β†’ ((𝑗 Β· π‘Œ) β‰  (0gβ€˜π‘Š) ↔ (𝑗 β‰  0 ∧ π‘Œ β‰  (0gβ€˜π‘Š))))
6358, 62mpbid 231 . . . . . . . 8 ((((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) ∧ (𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ))) β†’ (𝑗 β‰  0 ∧ π‘Œ β‰  (0gβ€˜π‘Š)))
6463simpld 495 . . . . . . 7 ((((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) ∧ (𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ))) β†’ 𝑗 β‰  0 )
65 eldifsn 4790 . . . . . . 7 (𝑗 ∈ (𝐾 βˆ– { 0 }) ↔ (𝑗 ∈ 𝐾 ∧ 𝑗 β‰  0 ))
6651, 64, 65sylanbrc 583 . . . . . 6 ((((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) ∧ (𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ))) β†’ 𝑗 ∈ (𝐾 βˆ– { 0 }))
6750, 66, 53reximssdv 3172 . . . . 5 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) β†’ βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ))
6838, 67pm2.61dane 3029 . . . 4 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ))
6968ex 413 . . 3 (πœ‘ β†’ ((π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}) β†’ βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ)))
701adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ π‘Š ∈ LVec)
71 eldifi 4126 . . . . . . . 8 (𝑗 ∈ (𝐾 βˆ– { 0 }) β†’ 𝑗 ∈ 𝐾)
7271adantl 482 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ 𝑗 ∈ 𝐾)
73 eldifsni 4793 . . . . . . . 8 (𝑗 ∈ (𝐾 βˆ– { 0 }) β†’ 𝑗 β‰  0 )
7473adantl 482 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ 𝑗 β‰  0 )
7530adantr 481 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ π‘Œ ∈ 𝑉)
7617, 4, 20, 6, 11, 26lspsnvs 20872 . . . . . . 7 ((π‘Š ∈ LVec ∧ (𝑗 ∈ 𝐾 ∧ 𝑗 β‰  0 ) ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{(𝑗 Β· π‘Œ)}) = (π‘β€˜{π‘Œ}))
7770, 72, 74, 75, 76syl121anc 1375 . . . . . 6 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ (π‘β€˜{(𝑗 Β· π‘Œ)}) = (π‘β€˜{π‘Œ}))
7877ex 413 . . . . 5 (πœ‘ β†’ (𝑗 ∈ (𝐾 βˆ– { 0 }) β†’ (π‘β€˜{(𝑗 Β· π‘Œ)}) = (π‘β€˜{π‘Œ})))
79 sneq 4638 . . . . . . 7 (𝑋 = (𝑗 Β· π‘Œ) β†’ {𝑋} = {(𝑗 Β· π‘Œ)})
8079fveqeq2d 6899 . . . . . 6 (𝑋 = (𝑗 Β· π‘Œ) β†’ ((π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}) ↔ (π‘β€˜{(𝑗 Β· π‘Œ)}) = (π‘β€˜{π‘Œ})))
8180biimprcd 249 . . . . 5 ((π‘β€˜{(𝑗 Β· π‘Œ)}) = (π‘β€˜{π‘Œ}) β†’ (𝑋 = (𝑗 Β· π‘Œ) β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})))
8278, 81syl6 35 . . . 4 (πœ‘ β†’ (𝑗 ∈ (𝐾 βˆ– { 0 }) β†’ (𝑋 = (𝑗 Β· π‘Œ) β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}))))
8382rexlimdv 3153 . . 3 (πœ‘ β†’ (βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ) β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})))
8469, 83impbid 211 . 2 (πœ‘ β†’ ((π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ)))
85 oveq1 7418 . . . 4 (𝑗 = π‘˜ β†’ (𝑗 Β· π‘Œ) = (π‘˜ Β· π‘Œ))
8685eqeq2d 2743 . . 3 (𝑗 = π‘˜ β†’ (𝑋 = (𝑗 Β· π‘Œ) ↔ 𝑋 = (π‘˜ Β· π‘Œ)))
8786cbvrexvw 3235 . 2 (βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ) ↔ βˆƒπ‘˜ ∈ (𝐾 βˆ– { 0 })𝑋 = (π‘˜ Β· π‘Œ))
8884, 87bitrdi 286 1 (πœ‘ β†’ ((π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ (𝐾 βˆ– { 0 })𝑋 = (π‘˜ Β· π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   βˆ– cdif 3945   βŠ† wss 3948  {csn 4628  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  Scalarcsca 17204   ·𝑠 cvsca 17205  0gc0g 17389  1rcur 20075  Ringcrg 20127  DivRingcdr 20500  LModclmod 20614  LSubSpclss 20686  LSpanclspn 20726  LVecclvec 20857
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  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 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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  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-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  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-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  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-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-sbg 18860  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-oppr 20225  df-dvdsr 20248  df-unit 20249  df-invr 20279  df-drng 20502  df-lmod 20616  df-lss 20687  df-lsp 20727  df-lvec 20858
This theorem is referenced by:  lspsneu  20881  mapdpglem26  40872  mapdpglem27  40873  hdmap14lem2a  41041  hdmap14lem2N  41043  prjsprellsp  41655
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