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Theorem lmhmlnmsplit 38177
Description: If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-2015.)
Hypotheses
Ref Expression
lmhmfgsplit.z 0 = (0g𝑇)
lmhmfgsplit.k 𝐾 = (𝐹 “ { 0 })
lmhmfgsplit.u 𝑈 = (𝑆s 𝐾)
lmhmfgsplit.v 𝑉 = (𝑇s ran 𝐹)
Assertion
Ref Expression
lmhmlnmsplit ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)

Proof of Theorem lmhmlnmsplit
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 lmhmlmod1 19255 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
213ad2ant1 1128 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LMod)
3 eqid 2760 . . . . . 6 (LSubSp‘𝑆) = (LSubSp‘𝑆)
4 eqid 2760 . . . . . 6 (𝑆s 𝑎) = (𝑆s 𝑎)
53, 4reslmhm 19274 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇))
653ad2antl1 1201 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇))
7 cnvresima 5784 . . . . . . . 8 ((𝐹𝑎) “ { 0 }) = ((𝐹 “ { 0 }) ∩ 𝑎)
8 lmhmfgsplit.k . . . . . . . . . 10 𝐾 = (𝐹 “ { 0 })
98eqcomi 2769 . . . . . . . . 9 (𝐹 “ { 0 }) = 𝐾
109ineq1i 3953 . . . . . . . 8 ((𝐹 “ { 0 }) ∩ 𝑎) = (𝐾𝑎)
11 incom 3948 . . . . . . . 8 (𝐾𝑎) = (𝑎𝐾)
127, 10, 113eqtri 2786 . . . . . . 7 ((𝐹𝑎) “ { 0 }) = (𝑎𝐾)
1312oveq2i 6825 . . . . . 6 ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = ((𝑆s 𝑎) ↾s (𝑎𝐾))
14 lmhmfgsplit.u . . . . . . . . 9 𝑈 = (𝑆s 𝐾)
1514oveq1i 6824 . . . . . . . 8 (𝑈s (𝑎𝐾)) = ((𝑆s 𝐾) ↾s (𝑎𝐾))
16 simpl1 1228 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
17 cnvexg 7278 . . . . . . . . . . . 12 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ V)
18 imaexg 7269 . . . . . . . . . . . 12 (𝐹 ∈ V → (𝐹 “ { 0 }) ∈ V)
1917, 18syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 “ { 0 }) ∈ V)
208, 19syl5eqel 2843 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ V)
2116, 20syl 17 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐾 ∈ V)
22 inss2 3977 . . . . . . . . 9 (𝑎𝐾) ⊆ 𝐾
23 ressabs 16161 . . . . . . . . 9 ((𝐾 ∈ V ∧ (𝑎𝐾) ⊆ 𝐾) → ((𝑆s 𝐾) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
2421, 22, 23sylancl 697 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝐾) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
2515, 24syl5eq 2806 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
26 vex 3343 . . . . . . . 8 𝑎 ∈ V
27 inss1 3976 . . . . . . . 8 (𝑎𝐾) ⊆ 𝑎
28 ressabs 16161 . . . . . . . 8 ((𝑎 ∈ V ∧ (𝑎𝐾) ⊆ 𝑎) → ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
2926, 27, 28mp2an 710 . . . . . . 7 ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾))
3025, 29syl6reqr 2813 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑈s (𝑎𝐾)))
3113, 30syl5eq 2806 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = (𝑈s (𝑎𝐾)))
32 simpl2 1230 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑈 ∈ LNoeM)
332adantr 472 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑆 ∈ LMod)
34 simpr 479 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑎 ∈ (LSubSp‘𝑆))
35 lmhmfgsplit.z . . . . . . . . . 10 0 = (0g𝑇)
368, 35, 3lmhmkerlss 19273 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ (LSubSp‘𝑆))
3716, 36syl 17 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐾 ∈ (LSubSp‘𝑆))
383lssincl 19187 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (LSubSp‘𝑆) ∧ 𝐾 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ∈ (LSubSp‘𝑆))
3933, 34, 37, 38syl3anc 1477 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ∈ (LSubSp‘𝑆))
4022a1i 11 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ⊆ 𝐾)
41 eqid 2760 . . . . . . . . 9 (LSubSp‘𝑈) = (LSubSp‘𝑈)
4214, 3, 41lsslss 19183 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝑆)) → ((𝑎𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎𝐾) ⊆ 𝐾)))
4333, 37, 42syl2anc 696 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑎𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎𝐾) ⊆ 𝐾)))
4439, 40, 43mpbir2and 995 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ∈ (LSubSp‘𝑈))
45 eqid 2760 . . . . . . 7 (𝑈s (𝑎𝐾)) = (𝑈s (𝑎𝐾))
4641, 45lnmlssfg 38170 . . . . . 6 ((𝑈 ∈ LNoeM ∧ (𝑎𝐾) ∈ (LSubSp‘𝑈)) → (𝑈s (𝑎𝐾)) ∈ LFinGen)
4732, 44, 46syl2anc 696 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈s (𝑎𝐾)) ∈ LFinGen)
4831, 47eqeltrd 2839 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) ∈ LFinGen)
49 lmhmfgsplit.v . . . . . . . . 9 𝑉 = (𝑇s ran 𝐹)
5049oveq1i 6824 . . . . . . . 8 (𝑉s ran (𝐹𝑎)) = ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎))
51 rnexg 7264 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ V)
52 resexg 5600 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹𝑎) ∈ V)
53 rnexg 7264 . . . . . . . . . 10 ((𝐹𝑎) ∈ V → ran (𝐹𝑎) ∈ V)
5452, 53syl 17 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran (𝐹𝑎) ∈ V)
55 ressress 16160 . . . . . . . . 9 ((ran 𝐹 ∈ V ∧ ran (𝐹𝑎) ∈ V) → ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
5651, 54, 55syl2anc 696 . . . . . . . 8 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
5750, 56syl5eq 2806 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑉s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
58 incom 3948 . . . . . . . . 9 (ran 𝐹 ∩ ran (𝐹𝑎)) = (ran (𝐹𝑎) ∩ ran 𝐹)
59 resss 5580 . . . . . . . . . . 11 (𝐹𝑎) ⊆ 𝐹
60 rnss 5509 . . . . . . . . . . 11 ((𝐹𝑎) ⊆ 𝐹 → ran (𝐹𝑎) ⊆ ran 𝐹)
6159, 60ax-mp 5 . . . . . . . . . 10 ran (𝐹𝑎) ⊆ ran 𝐹
62 df-ss 3729 . . . . . . . . . 10 (ran (𝐹𝑎) ⊆ ran 𝐹 ↔ (ran (𝐹𝑎) ∩ ran 𝐹) = ran (𝐹𝑎))
6361, 62mpbi 220 . . . . . . . . 9 (ran (𝐹𝑎) ∩ ran 𝐹) = ran (𝐹𝑎)
6458, 63eqtr2i 2783 . . . . . . . 8 ran (𝐹𝑎) = (ran 𝐹 ∩ ran (𝐹𝑎))
6564oveq2i 6825 . . . . . . 7 (𝑇s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎)))
6657, 65syl6reqr 2813 . . . . . 6 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑇s ran (𝐹𝑎)) = (𝑉s ran (𝐹𝑎)))
6716, 66syl 17 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇s ran (𝐹𝑎)) = (𝑉s ran (𝐹𝑎)))
68 simpl3 1232 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑉 ∈ LNoeM)
69 lmhmrnlss 19272 . . . . . . . 8 ((𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇) → ran (𝐹𝑎) ∈ (LSubSp‘𝑇))
706, 69syl 17 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹𝑎) ∈ (LSubSp‘𝑇))
7161a1i 11 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹𝑎) ⊆ ran 𝐹)
72 lmhmlmod2 19254 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7316, 72syl 17 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑇 ∈ LMod)
74 lmhmrnlss 19272 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))
7516, 74syl 17 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran 𝐹 ∈ (LSubSp‘𝑇))
76 eqid 2760 . . . . . . . . 9 (LSubSp‘𝑇) = (LSubSp‘𝑇)
77 eqid 2760 . . . . . . . . 9 (LSubSp‘𝑉) = (LSubSp‘𝑉)
7849, 76, 77lsslss 19183 . . . . . . . 8 ((𝑇 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑇)) → (ran (𝐹𝑎) ∈ (LSubSp‘𝑉) ↔ (ran (𝐹𝑎) ∈ (LSubSp‘𝑇) ∧ ran (𝐹𝑎) ⊆ ran 𝐹)))
7973, 75, 78syl2anc 696 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (ran (𝐹𝑎) ∈ (LSubSp‘𝑉) ↔ (ran (𝐹𝑎) ∈ (LSubSp‘𝑇) ∧ ran (𝐹𝑎) ⊆ ran 𝐹)))
8070, 71, 79mpbir2and 995 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹𝑎) ∈ (LSubSp‘𝑉))
81 eqid 2760 . . . . . . 7 (𝑉s ran (𝐹𝑎)) = (𝑉s ran (𝐹𝑎))
8277, 81lnmlssfg 38170 . . . . . 6 ((𝑉 ∈ LNoeM ∧ ran (𝐹𝑎) ∈ (LSubSp‘𝑉)) → (𝑉s ran (𝐹𝑎)) ∈ LFinGen)
8368, 80, 82syl2anc 696 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑉s ran (𝐹𝑎)) ∈ LFinGen)
8467, 83eqeltrd 2839 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇s ran (𝐹𝑎)) ∈ LFinGen)
85 eqid 2760 . . . . 5 ((𝐹𝑎) “ { 0 }) = ((𝐹𝑎) “ { 0 })
86 eqid 2760 . . . . 5 ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 }))
87 eqid 2760 . . . . 5 (𝑇s ran (𝐹𝑎)) = (𝑇s ran (𝐹𝑎))
8835, 85, 86, 87lmhmfgsplit 38176 . . . 4 (((𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇) ∧ ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) ∈ LFinGen ∧ (𝑇s ran (𝐹𝑎)) ∈ LFinGen) → (𝑆s 𝑎) ∈ LFinGen)
896, 48, 84, 88syl3anc 1477 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑆s 𝑎) ∈ LFinGen)
9089ralrimiva 3104 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆s 𝑎) ∈ LFinGen)
913islnm 38167 . 2 (𝑆 ∈ LNoeM ↔ (𝑆 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆s 𝑎) ∈ LFinGen))
922, 90, 91sylanbrc 701 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cin 3714  wss 3715  {csn 4321  ccnv 5265  ran crn 5267  cres 5268  cima 5269  cfv 6049  (class class class)co 6814  s cress 16080  0gc0g 16322  LModclmod 19085  LSubSpclss 19154   LMHom clmhm 19241  LFinGenclfig 38157  LNoeMclnm 38165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-sca 16179  df-vsca 16180  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-submnd 17557  df-grp 17646  df-minusg 17647  df-sbg 17648  df-subg 17812  df-ghm 17879  df-cntz 17970  df-lsm 18271  df-cmn 18415  df-abl 18416  df-mgp 18710  df-ur 18722  df-ring 18769  df-lmod 19087  df-lss 19155  df-lsp 19194  df-lmhm 19244  df-lfig 38158  df-lnm 38166
This theorem is referenced by:  pwslnmlem2  38183
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