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Theorem lmhmlnmsplit 37137
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 18952 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
213ad2ant1 1080 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LMod)
3 eqid 2621 . . . . . 6 (LSubSp‘𝑆) = (LSubSp‘𝑆)
4 eqid 2621 . . . . . 6 (𝑆s 𝑎) = (𝑆s 𝑎)
53, 4reslmhm 18971 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇))
653ad2antl1 1221 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇))
7 cnvresima 5582 . . . . . . . 8 ((𝐹𝑎) “ { 0 }) = ((𝐹 “ { 0 }) ∩ 𝑎)
8 lmhmfgsplit.k . . . . . . . . . 10 𝐾 = (𝐹 “ { 0 })
98eqcomi 2630 . . . . . . . . 9 (𝐹 “ { 0 }) = 𝐾
109ineq1i 3788 . . . . . . . 8 ((𝐹 “ { 0 }) ∩ 𝑎) = (𝐾𝑎)
11 incom 3783 . . . . . . . 8 (𝐾𝑎) = (𝑎𝐾)
127, 10, 113eqtri 2647 . . . . . . 7 ((𝐹𝑎) “ { 0 }) = (𝑎𝐾)
1312oveq2i 6615 . . . . . 6 ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = ((𝑆s 𝑎) ↾s (𝑎𝐾))
14 lmhmfgsplit.u . . . . . . . . 9 𝑈 = (𝑆s 𝐾)
1514oveq1i 6614 . . . . . . . 8 (𝑈s (𝑎𝐾)) = ((𝑆s 𝐾) ↾s (𝑎𝐾))
16 simpl1 1062 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
17 cnvexg 7059 . . . . . . . . . . . 12 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ V)
18 imaexg 7050 . . . . . . . . . . . 12 (𝐹 ∈ V → (𝐹 “ { 0 }) ∈ V)
1917, 18syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 “ { 0 }) ∈ V)
208, 19syl5eqel 2702 . . . . . . . . . 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 3812 . . . . . . . . 9 (𝑎𝐾) ⊆ 𝐾
23 ressabs 15860 . . . . . . . . 9 ((𝐾 ∈ V ∧ (𝑎𝐾) ⊆ 𝐾) → ((𝑆s 𝐾) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
2421, 22, 23sylancl 693 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝐾) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
2515, 24syl5eq 2667 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
26 vex 3189 . . . . . . . 8 𝑎 ∈ V
27 inss1 3811 . . . . . . . 8 (𝑎𝐾) ⊆ 𝑎
28 ressabs 15860 . . . . . . . 8 ((𝑎 ∈ V ∧ (𝑎𝐾) ⊆ 𝑎) → ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
2926, 27, 28mp2an 707 . . . . . . 7 ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾))
3025, 29syl6reqr 2674 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑈s (𝑎𝐾)))
3113, 30syl5eq 2667 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = (𝑈s (𝑎𝐾)))
32 simpl2 1063 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑈 ∈ LNoeM)
332adantr 481 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑆 ∈ LMod)
34 simpr 477 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑎 ∈ (LSubSp‘𝑆))
35 lmhmfgsplit.z . . . . . . . . . 10 0 = (0g𝑇)
368, 35, 3lmhmkerlss 18970 . . . . . . . . 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 18884 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (LSubSp‘𝑆) ∧ 𝐾 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ∈ (LSubSp‘𝑆))
3933, 34, 37, 38syl3anc 1323 . . . . . . 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 2621 . . . . . . . . 9 (LSubSp‘𝑈) = (LSubSp‘𝑈)
4214, 3, 41lsslss 18880 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝑆)) → ((𝑎𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎𝐾) ⊆ 𝐾)))
4333, 37, 42syl2anc 692 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑎𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎𝐾) ⊆ 𝐾)))
4439, 40, 43mpbir2and 956 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ∈ (LSubSp‘𝑈))
45 eqid 2621 . . . . . . 7 (𝑈s (𝑎𝐾)) = (𝑈s (𝑎𝐾))
4641, 45lnmlssfg 37130 . . . . . 6 ((𝑈 ∈ LNoeM ∧ (𝑎𝐾) ∈ (LSubSp‘𝑈)) → (𝑈s (𝑎𝐾)) ∈ LFinGen)
4732, 44, 46syl2anc 692 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈s (𝑎𝐾)) ∈ LFinGen)
4831, 47eqeltrd 2698 . . . 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 6614 . . . . . . . 8 (𝑉s ran (𝐹𝑎)) = ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎))
51 rnexg 7045 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ V)
52 resexg 5401 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹𝑎) ∈ V)
53 rnexg 7045 . . . . . . . . . 10 ((𝐹𝑎) ∈ V → ran (𝐹𝑎) ∈ V)
5452, 53syl 17 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran (𝐹𝑎) ∈ V)
55 ressress 15859 . . . . . . . . 9 ((ran 𝐹 ∈ V ∧ ran (𝐹𝑎) ∈ V) → ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
5651, 54, 55syl2anc 692 . . . . . . . 8 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
5750, 56syl5eq 2667 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑉s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
58 incom 3783 . . . . . . . . 9 (ran 𝐹 ∩ ran (𝐹𝑎)) = (ran (𝐹𝑎) ∩ ran 𝐹)
59 resss 5381 . . . . . . . . . . 11 (𝐹𝑎) ⊆ 𝐹
60 rnss 5314 . . . . . . . . . . 11 ((𝐹𝑎) ⊆ 𝐹 → ran (𝐹𝑎) ⊆ ran 𝐹)
6159, 60ax-mp 5 . . . . . . . . . 10 ran (𝐹𝑎) ⊆ ran 𝐹
62 df-ss 3569 . . . . . . . . . 10 (ran (𝐹𝑎) ⊆ ran 𝐹 ↔ (ran (𝐹𝑎) ∩ ran 𝐹) = ran (𝐹𝑎))
6361, 62mpbi 220 . . . . . . . . 9 (ran (𝐹𝑎) ∩ ran 𝐹) = ran (𝐹𝑎)
6458, 63eqtr2i 2644 . . . . . . . 8 ran (𝐹𝑎) = (ran 𝐹 ∩ ran (𝐹𝑎))
6564oveq2i 6615 . . . . . . 7 (𝑇s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎)))
6657, 65syl6reqr 2674 . . . . . 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 1064 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑉 ∈ LNoeM)
69 lmhmrnlss 18969 . . . . . . . 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 18951 . . . . . . . . 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 18969 . . . . . . . . 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 2621 . . . . . . . . 9 (LSubSp‘𝑇) = (LSubSp‘𝑇)
77 eqid 2621 . . . . . . . . 9 (LSubSp‘𝑉) = (LSubSp‘𝑉)
7849, 76, 77lsslss 18880 . . . . . . . 8 ((𝑇 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑇)) → (ran (𝐹𝑎) ∈ (LSubSp‘𝑉) ↔ (ran (𝐹𝑎) ∈ (LSubSp‘𝑇) ∧ ran (𝐹𝑎) ⊆ ran 𝐹)))
7973, 75, 78syl2anc 692 . . . . . . 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 956 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹𝑎) ∈ (LSubSp‘𝑉))
81 eqid 2621 . . . . . . 7 (𝑉s ran (𝐹𝑎)) = (𝑉s ran (𝐹𝑎))
8277, 81lnmlssfg 37130 . . . . . 6 ((𝑉 ∈ LNoeM ∧ ran (𝐹𝑎) ∈ (LSubSp‘𝑉)) → (𝑉s ran (𝐹𝑎)) ∈ LFinGen)
8368, 80, 82syl2anc 692 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑉s ran (𝐹𝑎)) ∈ LFinGen)
8467, 83eqeltrd 2698 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇s ran (𝐹𝑎)) ∈ LFinGen)
85 eqid 2621 . . . . 5 ((𝐹𝑎) “ { 0 }) = ((𝐹𝑎) “ { 0 })
86 eqid 2621 . . . . 5 ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 }))
87 eqid 2621 . . . . 5 (𝑇s ran (𝐹𝑎)) = (𝑇s ran (𝐹𝑎))
8835, 85, 86, 87lmhmfgsplit 37136 . . . 4 (((𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇) ∧ ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) ∈ LFinGen ∧ (𝑇s ran (𝐹𝑎)) ∈ LFinGen) → (𝑆s 𝑎) ∈ LFinGen)
896, 48, 84, 88syl3anc 1323 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑆s 𝑎) ∈ LFinGen)
9089ralrimiva 2960 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆s 𝑎) ∈ LFinGen)
913islnm 37127 . 2 (𝑆 ∈ LNoeM ↔ (𝑆 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆s 𝑎) ∈ LFinGen))
922, 90, 91sylanbrc 697 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cin 3554  wss 3555  {csn 4148  ccnv 5073  ran crn 5075  cres 5076  cima 5077  cfv 5847  (class class class)co 6604  s cress 15782  0gc0g 16021  LModclmod 18784  LSubSpclss 18851   LMHom clmhm 18938  LFinGenclfig 37117  LNoeMclnm 37125
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-sca 15878  df-vsca 15879  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-subg 17512  df-ghm 17579  df-cntz 17671  df-lsm 17972  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-ring 18470  df-lmod 18786  df-lss 18852  df-lsp 18891  df-lmhm 18941  df-lfig 37118  df-lnm 37126
This theorem is referenced by:  pwslnmlem2  37143
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