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Theorem lsmsat 36024
Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 36821 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 2818 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2818 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 lsmsat.o . . . . 5 0 = (0g𝑊)
6 lsmsat.a . . . . 5 𝐴 = (LSAtoms‘𝑊)
73, 4, 5, 6islsat 36007 . . . 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 233 . 2 (𝜑 → ∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟}))
10 simp3 1130 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑄 = ((LSpan‘𝑊)‘{𝑟}))
11 lsmsat.l . . . . . . . . . 10 (𝜑𝑄 ⊆ (𝑇 𝑈))
12113ad2ant1 1125 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑄 ⊆ (𝑇 𝑈))
1310, 12eqsstrrd 4003 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑇 𝑈))
14 lsmsat.s . . . . . . . . 9 𝑆 = (LSubSp‘𝑊)
1523ad2ant1 1125 . . . . . . . . 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 19784 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) ∈ 𝑆)
202, 16, 17, 19syl3anc 1363 . . . . . . . . . 10 (𝜑 → (𝑇 𝑈) ∈ 𝑆)
21203ad2ant1 1125 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑇 𝑈) ∈ 𝑆)
22 eldifi 4100 . . . . . . . . . 10 (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑟 ∈ (Base‘𝑊))
23223ad2ant2 1126 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (Base‘𝑊))
243, 14, 4, 15, 21, 23lspsnel5 19696 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑟 ∈ (𝑇 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑇 𝑈)))
2513, 24mpbird 258 . . . . . . 7 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (𝑇 𝑈))
2614lsssssubg 19659 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
2715, 26syl 17 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑆 ⊆ (SubGrp‘𝑊))
28163ad2ant1 1125 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇𝑆)
2927, 28sseldd 3965 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇 ∈ (SubGrp‘𝑊))
30173ad2ant1 1125 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈𝑆)
3127, 30sseldd 3965 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈 ∈ (SubGrp‘𝑊))
32 eqid 2818 . . . . . . . . 9 (+g𝑊) = (+g𝑊)
3332, 18lsmelval 18703 . . . . . . . 8 ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑟 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧)))
3429, 31, 33syl2anc 584 . . . . . . 7 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑟 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧)))
3525, 34mpbid 233 . . . . . 6 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧))
36 lsmsat.n . . . . . . . . . . . . . . 15 (𝜑𝑇 ≠ { 0 })
375, 14lssne0 19651 . . . . . . . . . . . . . . . 16 (𝑇𝑆 → (𝑇 ≠ { 0 } ↔ ∃𝑞𝑇 𝑞0 ))
3816, 37syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑇 ≠ { 0 } ↔ ∃𝑞𝑇 𝑞0 ))
3936, 38mpbid 233 . . . . . . . . . . . . . 14 (𝜑 → ∃𝑞𝑇 𝑞0 )
4039adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → ∃𝑞𝑇 𝑞0 )
41403ad2ant1 1125 . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . . . . 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 1187 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑞𝑇)
503, 14lssel 19638 . . . . . . . . . . . . . . . . 17 ((𝑇𝑆𝑞𝑇) → 𝑞 ∈ (Base‘𝑊))
5148, 49, 50syl2anc 584 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑞 ∈ (Base‘𝑊))
52 simpr3 1188 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑞0 )
533, 4, 5, 6lsatlspsn2 36008 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊) ∧ 𝑞0 ) → ((LSpan‘𝑊)‘{𝑞}) ∈ 𝐴)
5445, 51, 52, 53syl3anc 1363 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑞}) ∈ 𝐴)
5514, 4, 45, 48, 49lspsnel5a 19697 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇)
56 simpl3 1185 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑟 = (𝑦(+g𝑊)𝑧))
57 simpr1 1186 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑦 = 0 )
5857oveq1d 7160 . . . . . . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑈𝑆)
61 simp2r 1192 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑧𝑈)
623, 14lssel 19638 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑈𝑆𝑧𝑈) → 𝑧 ∈ (Base‘𝑊))
6360, 61, 62syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 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 19593 . . . . . . . . . . . . . . . . . . . . 21 ((𝑊 ∈ LMod ∧ 𝑧 ∈ (Base‘𝑊)) → ( 0 (+g𝑊)𝑧) = 𝑧)
6645, 64, 65syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ( 0 (+g𝑊)𝑧) = 𝑧)
6756, 58, 663eqtrd 2857 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑟 = 𝑧)
6867sneqd 4569 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → {𝑟} = {𝑧})
6968fveq2d 6667 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑟}) = ((LSpan‘𝑊)‘{𝑧}))
7014, 4, 44, 60, 61lspsnel5a 19697 . . . . . . . . . . . . . . . . . 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 4002 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑟}) ⊆ 𝑈)
733, 4lspsnsubg 19681 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊))
7445, 51, 73syl2anc 584 . . . . . . . . . . . . . . . . 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 3965 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑈 ∈ (SubGrp‘𝑊))
7818lsmub2 18712 . . . . . . . . . . . . . . . . 17 ((((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))
7974, 77, 78syl2anc 584 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑈 ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))
8072, 79sstrd 3974 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))
81 sseq1 3989 . . . . . . . . . . . . . . . . 17 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝𝑇 ↔ ((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇))
82 oveq1 7152 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 𝑈) = (((LSpan‘𝑊)‘{𝑞}) 𝑈))
8382sseq2d 3996 . . . . . . . . . . . . . . . . 17 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈)))
8481, 83anbi12d 630 . . . . . . . . . . . . . . . 16 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → ((𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)) ↔ (((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))))
8584rspcev 3620 . . . . . . . . . . . . . . 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 832 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
87863exp2 1346 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → (𝑦 = 0 → (𝑞𝑇 → (𝑞0 → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))))
8887imp 407 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦 = 0 ) → (𝑞𝑇 → (𝑞0 → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))))
8988rexlimdv 3280 . . . . . . . . . . 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 1191 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑦𝑇)
933, 14lssel 19638 . . . . . . . . . . . . . 14 ((𝑇𝑆𝑦𝑇) → 𝑦 ∈ (Base‘𝑊))
9447, 92, 93syl2anc 584 . . . . . . . . . . . . 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 485 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → 𝑦0 )
973, 4, 5, 6lsatlspsn2 36008 . . . . . . . . . . . 12 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑦0 ) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝐴)
9891, 95, 96, 97syl3anc 1363 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝐴)
9914, 4, 44, 47, 92lspsnel5a 19697 . . . . . . . . . . . 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 1130 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑟 = (𝑦(+g𝑊)𝑧))
102101sneqd 4569 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → {𝑟} = {(𝑦(+g𝑊)𝑧)})
103102fveq2d 6667 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑟}) = ((LSpan‘𝑊)‘{(𝑦(+g𝑊)𝑧)}))
1043, 32, 4lspvadd 19797 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{(𝑦(+g𝑊)𝑧)}) ⊆ ((LSpan‘𝑊)‘{𝑦, 𝑧}))
10544, 94, 63, 104syl3anc 1363 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{(𝑦(+g𝑊)𝑧)}) ⊆ ((LSpan‘𝑊)‘{𝑦, 𝑧}))
106103, 105eqsstrd 4002 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑟}) ⊆ ((LSpan‘𝑊)‘{𝑦, 𝑧}))
1073, 4, 18, 44, 94, 63lsmpr 19790 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦, 𝑧}) = (((LSpan‘𝑊)‘{𝑦}) ((LSpan‘𝑊)‘{𝑧})))
108106, 107sseqtrd 4004 . . . . . . . . . . . . 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 19678 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
11144, 94, 110syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
112109, 111sseldd 3965 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ (SubGrp‘𝑊))
113109, 60sseldd 3965 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑈 ∈ (SubGrp‘𝑊))
11418lsmless2 18715 . . . . . . . . . . . . . 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 1363 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → (((LSpan‘𝑊)‘{𝑦}) ((LSpan‘𝑊)‘{𝑧})) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))
116108, 115sstrd 3974 . . . . . . . . . . . 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 3989 . . . . . . . . . . . . 13 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝𝑇 ↔ ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇))
119 oveq1 7152 . . . . . . . . . . . . . 14 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 𝑈) = (((LSpan‘𝑊)‘{𝑦}) 𝑈))
120119sseq2d 3996 . . . . . . . . . . . . 13 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈)))
121118, 120anbi12d 630 . . . . . . . . . . . 12 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → ((𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)) ↔ (((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))))
122121rspcev 3620 . . . . . . . . . . 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 832 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
12490, 123pm2.61dane 3101 . . . . . . . . 9 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
1251243exp 1111 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → ((𝑦𝑇𝑧𝑈) → (𝑟 = (𝑦(+g𝑊)𝑧) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))))
126125rexlimdvv 3290 . . . . . . 7 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → (∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
1271263adant3 1124 . . . . . 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 3989 . . . . . . . 8 (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (𝑄 ⊆ (𝑝 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
130129anbi2d 628 . . . . . . 7 (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ((𝑝𝑇𝑄 ⊆ (𝑝 𝑈)) ↔ (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
131130rexbidv 3294 . . . . . 6 (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)) ↔ ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
1321313ad2ant3 1127 . . . . 5 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)) ↔ ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
133128, 132mpbird 258 . . . 4 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)))
1341333exp 1111 . . 3 (𝜑 → (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)))))
135134rexlimdv 3280 . 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wrex 3136  cdif 3930  wss 3933  {csn 4557  {cpr 4559  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  SubGrpcsubg 18211  LSSumclsm 18688  LModclmod 19563  LSubSpclss 19632  LSpanclspn 19672  LSAtomsclsa 35990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-subg 18214  df-cntz 18385  df-lsm 18690  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-lmod 19565  df-lss 19633  df-lsp 19673  df-lsatoms 35992
This theorem is referenced by:  dochexmidlem4  38479
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