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Theorem lsmsat 34964
Description: Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 35761 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 2765 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2765 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 lsmsat.o . . . . 5 0 = (0g𝑊)
6 lsmsat.a . . . . 5 𝐴 = (LSAtoms‘𝑊)
73, 4, 5, 6islsat 34947 . . . 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 223 . 2 (𝜑 → ∃𝑟 ∈ ((Base‘𝑊) ∖ { 0 })𝑄 = ((LSpan‘𝑊)‘{𝑟}))
10 simp3 1168 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑄 = ((LSpan‘𝑊)‘{𝑟}))
11 lsmsat.l . . . . . . . . . 10 (𝜑𝑄 ⊆ (𝑇 𝑈))
12113ad2ant1 1163 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑄 ⊆ (𝑇 𝑈))
1310, 12eqsstr3d 3800 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑇 𝑈))
14 lsmsat.s . . . . . . . . 9 𝑆 = (LSubSp‘𝑊)
1523ad2ant1 1163 . . . . . . . . 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 19355 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) ∈ 𝑆)
202, 16, 17, 19syl3anc 1490 . . . . . . . . . 10 (𝜑 → (𝑇 𝑈) ∈ 𝑆)
21203ad2ant1 1163 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑇 𝑈) ∈ 𝑆)
22 eldifi 3894 . . . . . . . . . 10 (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑟 ∈ (Base‘𝑊))
23223ad2ant2 1164 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (Base‘𝑊))
243, 14, 4, 15, 21, 23lspsnel5 19267 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑟 ∈ (𝑇 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑇 𝑈)))
2513, 24mpbird 248 . . . . . . 7 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑟 ∈ (𝑇 𝑈))
2614lsssssubg 19230 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
2715, 26syl 17 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑆 ⊆ (SubGrp‘𝑊))
28163ad2ant1 1163 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇𝑆)
2927, 28sseldd 3762 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑇 ∈ (SubGrp‘𝑊))
30173ad2ant1 1163 . . . . . . . . 9 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈𝑆)
3127, 30sseldd 3762 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → 𝑈 ∈ (SubGrp‘𝑊))
32 eqid 2765 . . . . . . . . 9 (+g𝑊) = (+g𝑊)
3332, 18lsmelval 18328 . . . . . . . 8 ((𝑇 ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → (𝑟 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧)))
3429, 31, 33syl2anc 579 . . . . . . 7 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (𝑟 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧)))
3525, 34mpbid 223 . . . . . 6 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧))
36 lsmsat.n . . . . . . . . . . . . . . 15 (𝜑𝑇 ≠ { 0 })
375, 14lssne0 19220 . . . . . . . . . . . . . . . 16 (𝑇𝑆 → (𝑇 ≠ { 0 } ↔ ∃𝑞𝑇 𝑞0 ))
3816, 37syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑇 ≠ { 0 } ↔ ∃𝑞𝑇 𝑞0 ))
3936, 38mpbid 223 . . . . . . . . . . . . . 14 (𝜑 → ∃𝑞𝑇 𝑞0 )
4039adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → ∃𝑞𝑇 𝑞0 )
41403ad2ant1 1163 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ∃𝑞𝑇 𝑞0 )
4241adantr 472 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦 = 0 ) → ∃𝑞𝑇 𝑞0 )
432adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑊 ∈ LMod)
44433ad2ant1 1163 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑊 ∈ LMod)
4544adantr 472 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑊 ∈ LMod)
4616adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑇𝑆)
47463ad2ant1 1163 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑇𝑆)
4847adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑇𝑆)
49 simpr2 1250 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑞𝑇)
503, 14lssel 19207 . . . . . . . . . . . . . . . . 17 ((𝑇𝑆𝑞𝑇) → 𝑞 ∈ (Base‘𝑊))
5148, 49, 50syl2anc 579 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑞 ∈ (Base‘𝑊))
52 simpr3 1252 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑞0 )
533, 4, 5, 6lsatlspsn2 34948 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊) ∧ 𝑞0 ) → ((LSpan‘𝑊)‘{𝑞}) ∈ 𝐴)
5445, 51, 52, 53syl3anc 1490 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑞}) ∈ 𝐴)
5514, 4, 45, 48, 49lspsnel5a 19268 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇)
56 simpl3 1246 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑟 = (𝑦(+g𝑊)𝑧))
57 simpr1 1248 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑦 = 0 )
5857oveq1d 6857 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → (𝑦(+g𝑊)𝑧) = ( 0 (+g𝑊)𝑧))
5917adantr 472 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑈𝑆)
60593ad2ant1 1163 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑈𝑆)
61 simp2r 1257 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑧𝑈)
623, 14lssel 19207 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑈𝑆𝑧𝑈) → 𝑧 ∈ (Base‘𝑊))
6360, 61, 62syl2anc 579 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑧 ∈ (Base‘𝑊))
6463adantr 472 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑧 ∈ (Base‘𝑊))
653, 32, 5lmod0vlid 19162 . . . . . . . . . . . . . . . . . . . . 21 ((𝑊 ∈ LMod ∧ 𝑧 ∈ (Base‘𝑊)) → ( 0 (+g𝑊)𝑧) = 𝑧)
6645, 64, 65syl2anc 579 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ( 0 (+g𝑊)𝑧) = 𝑧)
6756, 58, 663eqtrd 2803 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑟 = 𝑧)
6867sneqd 4346 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → {𝑟} = {𝑧})
6968fveq2d 6379 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑟}) = ((LSpan‘𝑊)‘{𝑧}))
7014, 4, 44, 60, 61lspsnel5a 19268 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑧}) ⊆ 𝑈)
7170adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑧}) ⊆ 𝑈)
7269, 71eqsstrd 3799 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑟}) ⊆ 𝑈)
733, 4lspsnsubg 19252 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊))
7445, 51, 73syl2anc 579 . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑈𝑆)
7775, 76sseldd 3762 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑈 ∈ (SubGrp‘𝑊))
7818lsmub2 18336 . . . . . . . . . . . . . . . . 17 ((((LSpan‘𝑊)‘{𝑞}) ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → 𝑈 ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))
7974, 77, 78syl2anc 579 . . . . . . . . . . . . . . . 16 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → 𝑈 ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))
8072, 79sstrd 3771 . . . . . . . . . . . . . . 15 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))
81 sseq1 3786 . . . . . . . . . . . . . . . . 17 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝𝑇 ↔ ((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇))
82 oveq1 6849 . . . . . . . . . . . . . . . . . 18 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (𝑝 𝑈) = (((LSpan‘𝑊)‘{𝑞}) 𝑈))
8382sseq2d 3793 . . . . . . . . . . . . . . . . 17 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈)))
8481, 83anbi12d 624 . . . . . . . . . . . . . . . 16 (𝑝 = ((LSpan‘𝑊)‘{𝑞}) → ((𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)) ↔ (((LSpan‘𝑊)‘{𝑞}) ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑞}) 𝑈))))
8584rspcev 3461 . . . . . . . . . . . . . . 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 865 . . . . . . . . . . . . . 14 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ (𝑦 = 0𝑞𝑇𝑞0 )) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
87863exp2 1463 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → (𝑦 = 0 → (𝑞𝑇 → (𝑞0 → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))))
8887imp 395 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦 = 0 ) → (𝑞𝑇 → (𝑞0 → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))))
8988rexlimdv 3177 . . . . . . . . . . 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 472 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → 𝑊 ∈ LMod)
92 simp2l 1256 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑦𝑇)
933, 14lssel 19207 . . . . . . . . . . . . . 14 ((𝑇𝑆𝑦𝑇) → 𝑦 ∈ (Base‘𝑊))
9447, 92, 93syl2anc 579 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑦 ∈ (Base‘𝑊))
9594adantr 472 . . . . . . . . . . . 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 34948 . . . . . . . . . . . 12 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑦0 ) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝐴)
9891, 95, 96, 97syl3anc 1490 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝐴)
9914, 4, 44, 47, 92lspsnel5a 19268 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇)
10099adantr 472 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇)
101 simp3 1168 . . . . . . . . . . . . . . . . 17 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑟 = (𝑦(+g𝑊)𝑧))
102101sneqd 4346 . . . . . . . . . . . . . . . 16 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → {𝑟} = {(𝑦(+g𝑊)𝑧)})
103102fveq2d 6379 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑟}) = ((LSpan‘𝑊)‘{(𝑦(+g𝑊)𝑧)}))
1043, 32, 4lspvadd 19368 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{(𝑦(+g𝑊)𝑧)}) ⊆ ((LSpan‘𝑊)‘{𝑦, 𝑧}))
10544, 94, 63, 104syl3anc 1490 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{(𝑦(+g𝑊)𝑧)}) ⊆ ((LSpan‘𝑊)‘{𝑦, 𝑧}))
106103, 105eqsstrd 3799 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑟}) ⊆ ((LSpan‘𝑊)‘{𝑦, 𝑧}))
1073, 4, 18, 44, 94, 63lsmpr 19361 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦, 𝑧}) = (((LSpan‘𝑊)‘{𝑦}) ((LSpan‘𝑊)‘{𝑧})))
108106, 107sseqtrd 3801 . . . . . . . . . . . . 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 19249 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑊)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
11144, 94, 110syl2anc 579 . . . . . . . . . . . . . . 15 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ 𝑆)
112109, 111sseldd 3762 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑦}) ∈ (SubGrp‘𝑊))
113109, 60sseldd 3762 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → 𝑈 ∈ (SubGrp‘𝑊))
11418lsmless2 18339 . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → (((LSpan‘𝑊)‘{𝑦}) ((LSpan‘𝑊)‘{𝑧})) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))
116108, 115sstrd 3771 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))
117116adantr 472 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))
118 sseq1 3786 . . . . . . . . . . . . 13 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝𝑇 ↔ ((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇))
119 oveq1 6849 . . . . . . . . . . . . . 14 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (𝑝 𝑈) = (((LSpan‘𝑊)‘{𝑦}) 𝑈))
120119sseq2d 3793 . . . . . . . . . . . . 13 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → (((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈)))
121118, 120anbi12d 624 . . . . . . . . . . . 12 (𝑝 = ((LSpan‘𝑊)‘{𝑦}) → ((𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)) ↔ (((LSpan‘𝑊)‘{𝑦}) ⊆ 𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (((LSpan‘𝑊)‘{𝑦}) 𝑈))))
122121rspcev 3461 . . . . . . . . . . 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 865 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) ∧ 𝑦0 ) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
12490, 123pm2.61dane 3024 . . . . . . . . 9 (((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) ∧ (𝑦𝑇𝑧𝑈) ∧ 𝑟 = (𝑦(+g𝑊)𝑧)) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
1251243exp 1148 . . . . . . . 8 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → ((𝑦𝑇𝑧𝑈) → (𝑟 = (𝑦(+g𝑊)𝑧) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))))
126125rexlimdvv 3184 . . . . . . 7 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 })) → (∃𝑦𝑇𝑧𝑈 𝑟 = (𝑦(+g𝑊)𝑧) → ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
1271263adant3 1162 . . . . . 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 3786 . . . . . . . 8 (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (𝑄 ⊆ (𝑝 𝑈) ↔ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈)))
130129anbi2d 622 . . . . . . 7 (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ((𝑝𝑇𝑄 ⊆ (𝑝 𝑈)) ↔ (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
131130rexbidv 3199 . . . . . 6 (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → (∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)) ↔ ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
1321313ad2ant3 1165 . . . . 5 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → (∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)) ↔ ∃𝑝𝐴 (𝑝𝑇 ∧ ((LSpan‘𝑊)‘{𝑟}) ⊆ (𝑝 𝑈))))
133128, 132mpbird 248 . . . 4 ((𝜑𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑄 = ((LSpan‘𝑊)‘{𝑟})) → ∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)))
1341333exp 1148 . . 3 (𝜑 → (𝑟 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑄 = ((LSpan‘𝑊)‘{𝑟}) → ∃𝑝𝐴 (𝑝𝑇𝑄 ⊆ (𝑝 𝑈)))))
135134rexlimdv 3177 . 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wrex 3056  cdif 3729  wss 3732  {csn 4334  {cpr 4336  cfv 6068  (class class class)co 6842  Basecbs 16130  +gcplusg 16214  0gc0g 16366  SubGrpcsubg 17852  LSSumclsm 18313  LModclmod 19132  LSubSpclss 19201  LSpanclspn 19243  LSAtomsclsa 34930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-0g 16368  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-submnd 17602  df-grp 17692  df-minusg 17693  df-sbg 17694  df-subg 17855  df-cntz 18013  df-lsm 18315  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-lmod 19134  df-lss 19202  df-lsp 19244  df-lsatoms 34932
This theorem is referenced by:  dochexmidlem4  37419
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