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Theorem lspsneq 20735
Description: Equal spans of singletons must have proportional vectors. See lspsnss2 20616 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 20717 . . . . . . . . . 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 20479 . . . . . . . . 9 (π‘Š ∈ LMod β†’ 𝑆 ∈ Ring)
6 lspsneq.k . . . . . . . . . 10 𝐾 = (Baseβ€˜π‘†)
7 eqid 2733 . . . . . . . . . 10 (1rβ€˜π‘†) = (1rβ€˜π‘†)
86, 7ringidcl 20083 . . . . . . . . 9 (𝑆 ∈ Ring β†’ (1rβ€˜π‘†) ∈ 𝐾)
93, 5, 83syl 18 . . . . . . . 8 (πœ‘ β†’ (1rβ€˜π‘†) ∈ 𝐾)
104lvecdrng 20716 . . . . . . . . 9 (π‘Š ∈ LVec β†’ 𝑆 ∈ DivRing)
11 lspsneq.o . . . . . . . . . 10 0 = (0gβ€˜π‘†)
1211, 7drngunz 20376 . . . . . . . . 9 (𝑆 ∈ DivRing β†’ (1rβ€˜π‘†) β‰  0 )
131, 10, 123syl 18 . . . . . . . 8 (πœ‘ β†’ (1rβ€˜π‘†) β‰  0 )
14 eldifsn 4791 . . . . . . . 8 ((1rβ€˜π‘†) ∈ (𝐾 βˆ– { 0 }) ↔ ((1rβ€˜π‘†) ∈ 𝐾 ∧ (1rβ€˜π‘†) β‰  0 ))
159, 13, 14sylanbrc 584 . . . . . . 7 (πœ‘ β†’ (1rβ€˜π‘†) ∈ (𝐾 βˆ– { 0 }))
1615ad2antrr 725 . . . . . 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 2733 . . . . . . . . . 10 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
1917, 18lmod0vcl 20501 . . . . . . . . 9 (π‘Š ∈ LMod β†’ (0gβ€˜π‘Š) ∈ 𝑉)
20 lspsneq.t . . . . . . . . . 10 Β· = ( ·𝑠 β€˜π‘Š)
2117, 4, 20, 7lmodvs1 20500 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (0gβ€˜π‘Š) ∈ 𝑉) β†’ ((1rβ€˜π‘†) Β· (0gβ€˜π‘Š)) = (0gβ€˜π‘Š))
223, 19, 21syl2anc2 586 . . . . . . . 8 (πœ‘ β†’ ((1rβ€˜π‘†) Β· (0gβ€˜π‘Š)) = (0gβ€˜π‘Š))
2322ad2antrr 725 . . . . . . 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 7417 . . . . . . . 8 (π‘Œ = (0gβ€˜π‘Š) β†’ ((1rβ€˜π‘†) Β· π‘Œ) = ((1rβ€˜π‘†) Β· (0gβ€˜π‘Š)))
2524adantl 483 . . . . . . 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 482 . . . . . . . . 9 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ π‘Š ∈ LMod)
28 lspsneq.x . . . . . . . . . 10 (πœ‘ β†’ 𝑋 ∈ 𝑉)
2928adantr 482 . . . . . . . . 9 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ 𝑋 ∈ 𝑉)
30 lspsneq.y . . . . . . . . . 10 (πœ‘ β†’ π‘Œ ∈ 𝑉)
3130adantr 482 . . . . . . . . 9 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ π‘Œ ∈ 𝑉)
32 simpr 486 . . . . . . . . 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 20624 . . . . . . . 8 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (𝑋 = (0gβ€˜π‘Š) ↔ π‘Œ = (0gβ€˜π‘Š)))
3433biimpar 479 . . . . . . 7 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ = (0gβ€˜π‘Š)) β†’ 𝑋 = (0gβ€˜π‘Š))
3523, 25, 343eqtr4rd 2784 . . . . . 6 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ = (0gβ€˜π‘Š)) β†’ 𝑋 = ((1rβ€˜π‘†) Β· π‘Œ))
36 oveq1 7416 . . . . . . 7 (𝑗 = (1rβ€˜π‘†) β†’ (𝑗 Β· π‘Œ) = ((1rβ€˜π‘†) Β· π‘Œ))
3736rspceeqv 3634 . . . . . 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 585 . . . . 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 4041 . . . . . . . . . 10 ((π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}) β†’ (π‘β€˜{𝑋}) βŠ† (π‘β€˜{π‘Œ}))
4039adantl 483 . . . . . . . . 9 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (π‘β€˜{𝑋}) βŠ† (π‘β€˜{π‘Œ}))
41 eqid 2733 . . . . . . . . . 10 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
4217, 41, 26lspsncl 20588 . . . . . . . . . . . 12 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
433, 30, 42syl2anc 585 . . . . . . . . . . 11 (πœ‘ β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
4443adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (π‘β€˜{π‘Œ}) ∈ (LSubSpβ€˜π‘Š))
4517, 41, 26, 27, 44, 29lspsnel5 20606 . . . . . . . . 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 257 . . . . . . . 8 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ 𝑋 ∈ (π‘β€˜{π‘Œ}))
474, 6, 17, 20, 26lspsnel 20614 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 ∈ (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘— ∈ 𝐾 𝑋 = (𝑗 Β· π‘Œ)))
4827, 31, 47syl2anc 585 . . . . . . . 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 482 . . . . . 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 770 . . . . . . 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 486 . . . . . . . . . . 11 ((𝑗 ∈ 𝐾 ∧ 𝑋 = (𝑗 Β· π‘Œ)) β†’ 𝑋 = (𝑗 Β· π‘Œ))
5352adantl 483 . . . . . . . . . 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 2962 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ (π‘Œ β‰  (0gβ€˜π‘Š) β†’ 𝑋 β‰  (0gβ€˜π‘Š)))
5655imp 408 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) ∧ π‘Œ β‰  (0gβ€˜π‘Š)) β†’ 𝑋 β‰  (0gβ€˜π‘Š))
5756adantr 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 β‰  (0gβ€˜π‘Š))
5853, 57eqnetrrd 3010 . . . . . . . . 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 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ π‘Š ∈ LVec)
6059ad2antrr 725 . . . . . . . . . 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 725 . . . . . . . . . 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 20722 . . . . . . . . 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 496 . . . . . . 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 4791 . . . . . . 7 (𝑗 ∈ (𝐾 βˆ– { 0 }) ↔ (𝑗 ∈ 𝐾 ∧ 𝑗 β‰  0 ))
6651, 64, 65sylanbrc 584 . . . . . 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 3173 . . . . 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 3030 . . . 4 ((πœ‘ ∧ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ})) β†’ βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ))
6968ex 414 . . 3 (πœ‘ β†’ ((π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}) β†’ βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ)))
701adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ π‘Š ∈ LVec)
71 eldifi 4127 . . . . . . . 8 (𝑗 ∈ (𝐾 βˆ– { 0 }) β†’ 𝑗 ∈ 𝐾)
7271adantl 483 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ 𝑗 ∈ 𝐾)
73 eldifsni 4794 . . . . . . . 8 (𝑗 ∈ (𝐾 βˆ– { 0 }) β†’ 𝑗 β‰  0 )
7473adantl 483 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ 𝑗 β‰  0 )
7530adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ π‘Œ ∈ 𝑉)
7617, 4, 20, 6, 11, 26lspsnvs 20727 . . . . . . 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 1376 . . . . . 6 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ (π‘β€˜{(𝑗 Β· π‘Œ)}) = (π‘β€˜{π‘Œ}))
7877ex 414 . . . . 5 (πœ‘ β†’ (𝑗 ∈ (𝐾 βˆ– { 0 }) β†’ (π‘β€˜{(𝑗 Β· π‘Œ)}) = (π‘β€˜{π‘Œ})))
79 sneq 4639 . . . . . . 7 (𝑋 = (𝑗 Β· π‘Œ) β†’ {𝑋} = {(𝑗 Β· π‘Œ)})
8079fveqeq2d 6900 . . . . . 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 3154 . . 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 7416 . . . 4 (𝑗 = π‘˜ β†’ (𝑗 Β· π‘Œ) = (π‘˜ Β· π‘Œ))
8685eqeq2d 2744 . . 3 (𝑗 = π‘˜ β†’ (𝑋 = (𝑗 Β· π‘Œ) ↔ 𝑋 = (π‘˜ Β· π‘Œ)))
8786cbvrexvw 3236 . 2 (βˆƒπ‘— ∈ (𝐾 βˆ– { 0 })𝑋 = (𝑗 Β· π‘Œ) ↔ βˆƒπ‘˜ ∈ (𝐾 βˆ– { 0 })𝑋 = (π‘˜ Β· π‘Œ))
8884, 87bitrdi 287 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071   βˆ– cdif 3946   βŠ† wss 3949  {csn 4629  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Scalarcsca 17200   ·𝑠 cvsca 17201  0gc0g 17385  1rcur 20004  Ringcrg 20056  DivRingcdr 20357  LModclmod 20471  LSubSpclss 20542  LSpanclspn 20582  LVecclvec 20713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mgp 19988  df-ur 20005  df-ring 20058  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-invr 20202  df-drng 20359  df-lmod 20473  df-lss 20543  df-lsp 20583  df-lvec 20714
This theorem is referenced by:  lspsneu  20736  mapdpglem26  40569  mapdpglem27  40570  hdmap14lem2a  40738  hdmap14lem2N  40740  prjsprellsp  41353
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