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Theorem lspsneq 20727
Description: Equal spans of singletons must have proportional vectors. See lspsnss2 20608 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 20709 . . . . . . . . . 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 20471 . . . . . . . . 9 (π‘Š ∈ LMod β†’ 𝑆 ∈ Ring)
6 lspsneq.k . . . . . . . . . 10 𝐾 = (Baseβ€˜π‘†)
7 eqid 2732 . . . . . . . . . 10 (1rβ€˜π‘†) = (1rβ€˜π‘†)
86, 7ringidcl 20076 . . . . . . . . 9 (𝑆 ∈ Ring β†’ (1rβ€˜π‘†) ∈ 𝐾)
93, 5, 83syl 18 . . . . . . . 8 (πœ‘ β†’ (1rβ€˜π‘†) ∈ 𝐾)
104lvecdrng 20708 . . . . . . . . 9 (π‘Š ∈ LVec β†’ 𝑆 ∈ DivRing)
11 lspsneq.o . . . . . . . . . 10 0 = (0gβ€˜π‘†)
1211, 7drngunz 20326 . . . . . . . . 9 (𝑆 ∈ DivRing β†’ (1rβ€˜π‘†) β‰  0 )
131, 10, 123syl 18 . . . . . . . 8 (πœ‘ β†’ (1rβ€˜π‘†) β‰  0 )
14 eldifsn 4789 . . . . . . . 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 20493 . . . . . . . . 9 (π‘Š ∈ LMod β†’ (0gβ€˜π‘Š) ∈ 𝑉)
20 lspsneq.t . . . . . . . . . 10 Β· = ( ·𝑠 β€˜π‘Š)
2117, 4, 20, 7lmodvs1 20492 . . . . . . . . 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 7413 . . . . . . . 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 20616 . . . . . . . 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 7412 . . . . . . 7 (𝑗 = (1rβ€˜π‘†) β†’ (𝑗 Β· π‘Œ) = ((1rβ€˜π‘†) Β· π‘Œ))
3736rspceeqv 3632 . . . . . 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 4039 . . . . . . . . . 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 20580 . . . . . . . . . . . 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 20598 . . . . . . . . 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 20606 . . . . . . . . 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 20714 . . . . . . . . 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 4789 . . . . . . 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 4125 . . . . . . . 8 (𝑗 ∈ (𝐾 βˆ– { 0 }) β†’ 𝑗 ∈ 𝐾)
7271adantl 482 . . . . . . 7 ((πœ‘ ∧ 𝑗 ∈ (𝐾 βˆ– { 0 })) β†’ 𝑗 ∈ 𝐾)
73 eldifsni 4792 . . . . . . . 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 20719 . . . . . . 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 4637 . . . . . . 7 (𝑋 = (𝑗 Β· π‘Œ) β†’ {𝑋} = {(𝑗 Β· π‘Œ)})
8079fveqeq2d 6896 . . . . . 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 7412 . . . 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 3944   βŠ† wss 3947  {csn 4627  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  1rcur 19998  Ringcrg 20049  DivRingcdr 20307  LModclmod 20463  LSubSpclss 20534  LSpanclspn 20574  LVecclvec 20705
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-tpos 8207  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-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mgp 19982  df-ur 19999  df-ring 20051  df-oppr 20142  df-dvdsr 20163  df-unit 20164  df-invr 20194  df-drng 20309  df-lmod 20465  df-lss 20535  df-lsp 20575  df-lvec 20706
This theorem is referenced by:  lspsneu  20728  mapdpglem26  40557  mapdpglem27  40558  hdmap14lem2a  40726  hdmap14lem2N  40728  prjsprellsp  41349
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