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Theorem lsmsat 33761
Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 34557 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)
Hypotheses
Ref Expression
lsmsat.o 0 = (0g𝑊)
lsmsat.s 𝑆 = (LSubSp‘𝑊)
lsmsat.p = (LSSum‘𝑊)
lsmsat.a 𝐴 = (LSAtoms‘𝑊)
lsmsat.w (𝜑𝑊 ∈ LMod)
lsmsat.t (𝜑𝑇𝑆)
lsmsat.u (𝜑𝑈𝑆)
lsmsat.q (𝜑𝑄𝐴)
lsmsat.n (𝜑𝑇 ≠ { 0 })
lsmsat.l (𝜑𝑄 ⊆ (𝑇 𝑈))
Assertion
Ref Expression
lsmsat (𝜑 → ∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)))
Distinct variable groups:   𝐴,𝑝   ,𝑝   𝑄,𝑝   𝑇,𝑝   𝑈,𝑝   𝑊,𝑝
Allowed substitution hints:   𝜑(𝑝)   𝑆(𝑝)   0 (𝑝)

Proof of Theorem lsmsat
Dummy variables 𝑞 𝑟 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmsat.q . . 3 (𝜑𝑄𝐴)
2 lsmsat.w . . . 4 (𝜑𝑊 ∈ LMod)
3 eqid 2626 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2626 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 lsmsat.o . . . . 5 0 = (0g𝑊)
6 lsmsat.a . . . . 5 𝐴 = (LSAtoms‘𝑊)
73, 4, 5, 6islsat 33744 . . . 4 (𝑊 ∈ LMod → (𝑄𝐴 ↔ ∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟})))
82, 7syl 17 . . 3 (𝜑 → (𝑄𝐴 ↔ ∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟})))
91, 8mpbid 222 . 2 (𝜑 → ∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟}))
10 simp3 1061 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑄 = ((LSpan‘𝑊)‘{𝑟}))
11 lsmsat.l . . . . . . . . . 10 (𝜑𝑄 ⊆ (𝑇 𝑈))
12113ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑄 ⊆ (𝑇 𝑈))
1310, 12eqsstr3d 3624 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑇 𝑈))
14 lsmsat.s . . . . . . . . 9 𝑆 = (LSubSp‘𝑊)
1523ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑊 ∈ LMod)
16 lsmsat.t . . . . . . . . . . 11 (𝜑𝑇𝑆)
17 lsmsat.u . . . . . . . . . . 11 (𝜑𝑈𝑆)
18 lsmsat.p . . . . . . . . . . . 12 = (LSSum‘𝑊)
1914, 18lsmcl 18997 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) ∈ 𝑆)
202, 16, 17, 19syl3anc 1323 . . . . . . . . . 10 (𝜑 → (𝑇 𝑈) ∈ 𝑆)
21203ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑇 𝑈) ∈ 𝑆)
22 eldifi 3715 . . . . . . . . . 10 (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑟 ∈ (Base‘𝑊))
23223ad2ant2 1081 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (Base‘𝑊))
243, 14, 4, 15, 21, 23lspsnel5 18909 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑟 ∈ (𝑇 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑇 𝑈)))
2513, 24mpbird 247 . . . . . . 7 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (𝑇 𝑈))
2614lsssssubg 18872 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
2715, 26syl 17 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑆 ⊆ (SubGrp‘𝑊))
28163ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇𝑆)
2927, 28sseldd 3589 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇 ∈ (SubGrp‘𝑊))
30173ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈𝑆)
3127, 30sseldd 3589 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈 ∈ (SubGrp‘𝑊))
32 eqid 2626 . . . . . . . . 9 (+g𝑊) = (+g𝑊)
3332, 18lsmelval 17980 . . . . . . . 8 ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑟 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧)))
3429, 31, 33syl2anc 692 . . . . . . 7 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑟 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧)))
3525, 34mpbid 222 . . . . . 6 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧))
36 lsmsat.n . . . . . . . . . . . . . . 15 (𝜑𝑇 ≠ { 0 })
375, 14lssne0 18865 . . . . . . . . . . . . . . . 16 (𝑇𝑆 → (𝑇 ≠ { 0 } ↔ ∃𝑞𝑇 𝑞0 ))
3816, 37syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑇 ≠ { 0 } ↔ ∃𝑞𝑇 𝑞0 ))
3936, 38mpbid 222 . . . . . . . . . . . . . 14 (𝜑 → ∃𝑞𝑇 𝑞0 )
4039adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → ∃𝑞𝑇 𝑞0 )
41403ad2ant1 1080 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ∃𝑞𝑇 𝑞0 )
4241adantr 481 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦 = 0 ) → ∃𝑞𝑇 𝑞0 )
432adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑊 ∈ LMod)
44433ad2ant1 1080 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑊 ∈ LMod)
4544adantr 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑊 ∈ LMod)
4616adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑇𝑆)
47463ad2ant1 1080 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑇𝑆)
4847adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑇𝑆)
49 simpr2 1066 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑞𝑇)
503, 14lssel 18852 . . . . . . . . . . . . . . . . 17 ((𝑇𝑆𝑞𝑇) → 𝑞 ∈ (Base‘𝑊))
5148, 49, 50syl2anc 692 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑞 ∈ (Base‘𝑊))
52 simpr3 1067 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑞0 )
533, 4, 5, 6lsatlspsn2 33745 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊) ∧ 𝑞0 ) → ((LSpan‘𝑊)‘{𝑞}) ∈ 𝐴)
5445, 51, 52, 53syl3anc 1323 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑞}) ∈ 𝐴)
5514, 4, 45, 48, 49lspsnel5a 18910 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇)
56 simpl3 1064 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑟 = (𝑦(+g𝑊)𝑧))
57 simpr1 1065 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑦 = 0 )
5857oveq1d 6620 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → (𝑦(+g𝑊)𝑧) = ( 0 (+g𝑊)𝑧))
5917adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑈𝑆)
60593ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑈𝑆)
61 simp2r 1086 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑧𝑈)
623, 14lssel 18852 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑈𝑆𝑧𝑈) → 𝑧 ∈ (Base‘𝑊))
6360, 61, 62syl2anc 692 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑧 ∈ (Base‘𝑊))
6463adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑧 ∈ (Base‘𝑊))
653, 32, 5lmod0vlid 18809 . . . . . . . . . . . . . . . . . . . . 21 ((𝑊 ∈ LMod ∧ 𝑧 ∈ (Base‘𝑊)) → ( 0 (+g𝑊)𝑧) = 𝑧)
6645, 64, 65syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ( 0 (+g𝑊)𝑧) = 𝑧)
6756, 58, 663eqtrd 2664 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑟 = 𝑧)
6867sneqd 4165 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → {𝑟} = {𝑧})
6968fveq2d 6154 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑟}) = ((LSpan‘𝑊)‘{𝑧}))
7014, 4, 44, 60, 61lspsnel5a 18910 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑧}) ⊆ 𝑈)
7170adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑧}) ⊆ 𝑈)
7269, 71eqsstrd 3623 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑟}) ⊆ 𝑈)
733, 4lspsnsubg 18894 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊))
7445, 51, 73syl2anc 692 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊))
7545, 26syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑆 ⊆ (SubGrp‘𝑊))
7660adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑈𝑆)
7775, 76sseldd 3589 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑈 ∈ (SubGrp‘𝑊))
7818lsmub2 17988 . . . . . . . . . . . . . . . . 17 ((((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))
7974, 77, 78syl2anc 692 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑈 ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))
8072, 79sstrd 3598 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))
81 sseq1 3610 . . . . . . . . . . . . . . . . 17 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝𝑇 ↔ ((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇))
82 oveq1 6612 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 𝑈) = (((LSpan‘𝑊)‘{𝑞}) 𝑈))
8382sseq2d 3617 . . . . . . . . . . . . . . . . 17 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈)))
8481, 83anbi12d 746 . . . . . . . . . . . . . . . 16 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → ((𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)) ↔ (((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))))
8584rspcev 3300 . . . . . . . . . . . . . . 15 ((((LSpan‘𝑊)‘{𝑞}) ∈ 𝐴 ∧ (((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
8654, 55, 80, 85syl12anc 1321 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
87863exp2 1282 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → (𝑦 = 0 → (𝑞𝑇 → (𝑞0 → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))))
8887imp 445 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦 = 0 ) → (𝑞𝑇 → (𝑞0 → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))))
8988rexlimdv 3028 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦 = 0 ) → (∃𝑞𝑇 𝑞0 → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
9042, 89mpd 15 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦 = 0 ) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
9144adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → 𝑊 ∈ LMod)
92 simp2l 1085 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑦𝑇)
933, 14lssel 18852 . . . . . . . . . . . . . 14 ((𝑇𝑆𝑦𝑇) → 𝑦 ∈ (Base‘𝑊))
9447, 92, 93syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑦 ∈ (Base‘𝑊))
9594adantr 481 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → 𝑦 ∈ (Base‘𝑊))
96 simpr 477 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → 𝑦0 )
973, 4, 5, 6lsatlspsn2 33745 . . . . . . . . . . . 12 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑦0 ) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝐴)
9891, 95, 96, 97syl3anc 1323 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝐴)
9914, 4, 44, 47, 92lspsnel5a 18910 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇)
10099adantr 481 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇)
101 simp3 1061 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑟 = (𝑦(+g𝑊)𝑧))
102101sneqd 4165 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → {𝑟} = {(𝑦(+g𝑊)𝑧)})
103102fveq2d 6154 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑟}) = ((LSpan‘𝑊)‘{(𝑦(+g𝑊)𝑧)}))
1043, 32, 4lspvadd 19010 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{(𝑦(+g𝑊)𝑧)}) ⊆ ((LSpan‘𝑊)‘{𝑦, 𝑧}))
10544, 94, 63, 104syl3anc 1323 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{(𝑦(+g𝑊)𝑧)}) ⊆ ((LSpan‘𝑊)‘{𝑦, 𝑧}))
106103, 105eqsstrd 3623 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑟}) ⊆ ((LSpan‘𝑊)‘{𝑦, 𝑧}))
1073, 4, 18, 44, 94, 63lsmpr 19003 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦, 𝑧}) = (((LSpan‘𝑊)‘{𝑦}) ((LSpan‘𝑊)‘{𝑧})))
108106, 107sseqtrd 3625 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) ((LSpan‘𝑊)‘{𝑧})))
10944, 26syl 17 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑆 ⊆ (SubGrp‘𝑊))
1103, 14, 4lspsncl 18891 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
11144, 94, 110syl2anc 692 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
112109, 111sseldd 3589 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ (SubGrp‘𝑊))
113109, 60sseldd 3589 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑈 ∈ (SubGrp‘𝑊))
11418lsmless2 17991 . . . . . . . . . . . . . 14 ((((LSpan‘𝑊)‘{𝑦}) ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊) ∧ ((LSpan‘𝑊)‘{𝑧}) ⊆ 𝑈) → (((LSpan‘𝑊)‘{𝑦}) ((LSpan‘𝑊)‘{𝑧})) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))
115112, 113, 70, 114syl3anc 1323 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → (((LSpan‘𝑊)‘{𝑦}) ((LSpan‘𝑊)‘{𝑧})) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))
116108, 115sstrd 3598 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))
117116adantr 481 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))
118 sseq1 3610 . . . . . . . . . . . . 13 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝𝑇 ↔ ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇))
119 oveq1 6612 . . . . . . . . . . . . . 14 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 𝑈) = (((LSpan‘𝑊)‘{𝑦}) 𝑈))
120119sseq2d 3617 . . . . . . . . . . . . 13 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈)))
121118, 120anbi12d 746 . . . . . . . . . . . 12 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → ((𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)) ↔ (((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))))
122121rspcev 3300 . . . . . . . . . . 11 ((((LSpan‘𝑊)‘{𝑦}) ∈ 𝐴 ∧ (((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
12398, 100, 117, 122syl12anc 1321 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
12490, 123pm2.61dane 2883 . . . . . . . . 9 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
1251243exp 1261 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → ((𝑦𝑇𝑧𝑈) → (𝑟 = (𝑦(+g𝑊)𝑧) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))))
126125rexlimdvv 3035 . . . . . . 7 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → (∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
1271263adant3 1079 . . . . . 6 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
12835, 127mpd 15 . . . . 5 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
129 sseq1 3610 . . . . . . . 8 (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (𝑄 ⊆ (𝑝 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
130129anbi2d 739 . . . . . . 7 (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ((𝑝𝑇𝑄 ⊆ (𝑝 𝑈)) ↔ (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
131130rexbidv 3050 . . . . . 6 (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)) ↔ ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
1321313ad2ant3 1082 . . . . 5 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)) ↔ ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
133128, 132mpbird 247 . . . 4 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)))
1341333exp 1261 . . 3 (𝜑 → (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)))))
135134rexlimdv 3028 . 2 (𝜑 → (∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈))))
1369, 135mpd 15 1 (𝜑 → ∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wrex 2913  cdif 3557  wss 3560  {csn 4153  {cpr 4155  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  0gc0g 16016  SubGrpcsubg 17504  LSSumclsm 17965  LModclmod 18779  LSubSpclss 18846  LSpanclspn 18885  LSAtomsclsa 33727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-submnd 17252  df-grp 17341  df-minusg 17342  df-sbg 17343  df-subg 17507  df-cntz 17666  df-lsm 17967  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-ring 18465  df-lmod 18781  df-lss 18847  df-lsp 18886  df-lsatoms 33729
This theorem is referenced by:  dochexmidlem4  36218
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