Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lmhmlnmsplit Structured version   Visualization version   GIF version

Theorem lmhmlnmsplit 43075
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 21049 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
213ad2ant1 1132 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LMod)
3 eqid 2734 . . . . . 6 (LSubSp‘𝑆) = (LSubSp‘𝑆)
4 eqid 2734 . . . . . 6 (𝑆s 𝑎) = (𝑆s 𝑎)
53, 4reslmhm 21068 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇))
653ad2antl1 1184 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇))
7 cnvresima 6251 . . . . . . . 8 ((𝐹𝑎) “ { 0 }) = ((𝐹 “ { 0 }) ∩ 𝑎)
8 lmhmfgsplit.k . . . . . . . . . 10 𝐾 = (𝐹 “ { 0 })
98eqcomi 2743 . . . . . . . . 9 (𝐹 “ { 0 }) = 𝐾
109ineq1i 4223 . . . . . . . 8 ((𝐹 “ { 0 }) ∩ 𝑎) = (𝐾𝑎)
11 incom 4216 . . . . . . . 8 (𝐾𝑎) = (𝑎𝐾)
127, 10, 113eqtri 2766 . . . . . . 7 ((𝐹𝑎) “ { 0 }) = (𝑎𝐾)
1312oveq2i 7441 . . . . . 6 ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = ((𝑆s 𝑎) ↾s (𝑎𝐾))
14 vex 3481 . . . . . . . 8 𝑎 ∈ V
15 inss1 4244 . . . . . . . 8 (𝑎𝐾) ⊆ 𝑎
16 ressabs 17294 . . . . . . . 8 ((𝑎 ∈ V ∧ (𝑎𝐾) ⊆ 𝑎) → ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
1714, 15, 16mp2an 692 . . . . . . 7 ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾))
18 lmhmfgsplit.u . . . . . . . . 9 𝑈 = (𝑆s 𝐾)
1918oveq1i 7440 . . . . . . . 8 (𝑈s (𝑎𝐾)) = ((𝑆s 𝐾) ↾s (𝑎𝐾))
20 simpl1 1190 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
21 cnvexg 7946 . . . . . . . . . . . 12 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ V)
22 imaexg 7935 . . . . . . . . . . . 12 (𝐹 ∈ V → (𝐹 “ { 0 }) ∈ V)
2321, 22syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 “ { 0 }) ∈ V)
248, 23eqeltrid 2842 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ V)
2520, 24syl 17 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐾 ∈ V)
26 inss2 4245 . . . . . . . . 9 (𝑎𝐾) ⊆ 𝐾
27 ressabs 17294 . . . . . . . . 9 ((𝐾 ∈ V ∧ (𝑎𝐾) ⊆ 𝐾) → ((𝑆s 𝐾) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
2825, 26, 27sylancl 586 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝐾) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
2919, 28eqtrid 2786 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
3017, 29eqtr4id 2793 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑈s (𝑎𝐾)))
3113, 30eqtrid 2786 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = (𝑈s (𝑎𝐾)))
32 simpl2 1191 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑈 ∈ LNoeM)
332adantr 480 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑆 ∈ LMod)
34 simpr 484 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑎 ∈ (LSubSp‘𝑆))
35 lmhmfgsplit.z . . . . . . . . . 10 0 = (0g𝑇)
368, 35, 3lmhmkerlss 21067 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ (LSubSp‘𝑆))
3720, 36syl 17 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐾 ∈ (LSubSp‘𝑆))
383lssincl 20980 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (LSubSp‘𝑆) ∧ 𝐾 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ∈ (LSubSp‘𝑆))
3933, 34, 37, 38syl3anc 1370 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ∈ (LSubSp‘𝑆))
4026a1i 11 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ⊆ 𝐾)
41 eqid 2734 . . . . . . . . 9 (LSubSp‘𝑈) = (LSubSp‘𝑈)
4218, 3, 41lsslss 20976 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝑆)) → ((𝑎𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎𝐾) ⊆ 𝐾)))
4333, 37, 42syl2anc 584 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑎𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎𝐾) ⊆ 𝐾)))
4439, 40, 43mpbir2and 713 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ∈ (LSubSp‘𝑈))
45 eqid 2734 . . . . . . 7 (𝑈s (𝑎𝐾)) = (𝑈s (𝑎𝐾))
4641, 45lnmlssfg 43068 . . . . . 6 ((𝑈 ∈ LNoeM ∧ (𝑎𝐾) ∈ (LSubSp‘𝑈)) → (𝑈s (𝑎𝐾)) ∈ LFinGen)
4732, 44, 46syl2anc 584 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈s (𝑎𝐾)) ∈ LFinGen)
4831, 47eqeltrd 2838 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) ∈ LFinGen)
49 incom 4216 . . . . . . . . 9 (ran 𝐹 ∩ ran (𝐹𝑎)) = (ran (𝐹𝑎) ∩ ran 𝐹)
50 resss 6021 . . . . . . . . . . 11 (𝐹𝑎) ⊆ 𝐹
51 rnss 5952 . . . . . . . . . . 11 ((𝐹𝑎) ⊆ 𝐹 → ran (𝐹𝑎) ⊆ ran 𝐹)
5250, 51ax-mp 5 . . . . . . . . . 10 ran (𝐹𝑎) ⊆ ran 𝐹
53 dfss2 3980 . . . . . . . . . 10 (ran (𝐹𝑎) ⊆ ran 𝐹 ↔ (ran (𝐹𝑎) ∩ ran 𝐹) = ran (𝐹𝑎))
5452, 53mpbi 230 . . . . . . . . 9 (ran (𝐹𝑎) ∩ ran 𝐹) = ran (𝐹𝑎)
5549, 54eqtr2i 2763 . . . . . . . 8 ran (𝐹𝑎) = (ran 𝐹 ∩ ran (𝐹𝑎))
5655oveq2i 7441 . . . . . . 7 (𝑇s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎)))
57 lmhmfgsplit.v . . . . . . . . 9 𝑉 = (𝑇s ran 𝐹)
5857oveq1i 7440 . . . . . . . 8 (𝑉s ran (𝐹𝑎)) = ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎))
59 rnexg 7924 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ V)
60 resexg 6046 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹𝑎) ∈ V)
61 rnexg 7924 . . . . . . . . . 10 ((𝐹𝑎) ∈ V → ran (𝐹𝑎) ∈ V)
6260, 61syl 17 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran (𝐹𝑎) ∈ V)
63 ressress 17293 . . . . . . . . 9 ((ran 𝐹 ∈ V ∧ ran (𝐹𝑎) ∈ V) → ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
6459, 62, 63syl2anc 584 . . . . . . . 8 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
6558, 64eqtrid 2786 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑉s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
6656, 65eqtr4id 2793 . . . . . 6 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑇s ran (𝐹𝑎)) = (𝑉s ran (𝐹𝑎)))
6720, 66syl 17 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇s ran (𝐹𝑎)) = (𝑉s ran (𝐹𝑎)))
68 simpl3 1192 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑉 ∈ LNoeM)
69 lmhmrnlss 21066 . . . . . . . 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))
7152a1i 11 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹𝑎) ⊆ ran 𝐹)
72 lmhmlmod2 21048 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7320, 72syl 17 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑇 ∈ LMod)
74 lmhmrnlss 21066 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))
7520, 74syl 17 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran 𝐹 ∈ (LSubSp‘𝑇))
76 eqid 2734 . . . . . . . . 9 (LSubSp‘𝑇) = (LSubSp‘𝑇)
77 eqid 2734 . . . . . . . . 9 (LSubSp‘𝑉) = (LSubSp‘𝑉)
7857, 76, 77lsslss 20976 . . . . . . . 8 ((𝑇 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑇)) → (ran (𝐹𝑎) ∈ (LSubSp‘𝑉) ↔ (ran (𝐹𝑎) ∈ (LSubSp‘𝑇) ∧ ran (𝐹𝑎) ⊆ ran 𝐹)))
7973, 75, 78syl2anc 584 . . . . . . 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 713 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹𝑎) ∈ (LSubSp‘𝑉))
81 eqid 2734 . . . . . . 7 (𝑉s ran (𝐹𝑎)) = (𝑉s ran (𝐹𝑎))
8277, 81lnmlssfg 43068 . . . . . 6 ((𝑉 ∈ LNoeM ∧ ran (𝐹𝑎) ∈ (LSubSp‘𝑉)) → (𝑉s ran (𝐹𝑎)) ∈ LFinGen)
8368, 80, 82syl2anc 584 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑉s ran (𝐹𝑎)) ∈ LFinGen)
8467, 83eqeltrd 2838 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇s ran (𝐹𝑎)) ∈ LFinGen)
85 eqid 2734 . . . . 5 ((𝐹𝑎) “ { 0 }) = ((𝐹𝑎) “ { 0 })
86 eqid 2734 . . . . 5 ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 }))
87 eqid 2734 . . . . 5 (𝑇s ran (𝐹𝑎)) = (𝑇s ran (𝐹𝑎))
8835, 85, 86, 87lmhmfgsplit 43074 . . . 4 (((𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇) ∧ ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) ∈ LFinGen ∧ (𝑇s ran (𝐹𝑎)) ∈ LFinGen) → (𝑆s 𝑎) ∈ LFinGen)
896, 48, 84, 88syl3anc 1370 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑆s 𝑎) ∈ LFinGen)
9089ralrimiva 3143 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆s 𝑎) ∈ LFinGen)
913islnm 43065 . 2 (𝑆 ∈ LNoeM ↔ (𝑆 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆s 𝑎) ∈ LFinGen))
922, 90, 91sylanbrc 583 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  cin 3961  wss 3962  {csn 4630  ccnv 5687  ran crn 5689  cres 5690  cima 5691  cfv 6562  (class class class)co 7430  s cress 17273  0gc0g 17485  LModclmod 20874  LSubSpclss 20946   LMHom clmhm 21035  LFinGenclfig 43055  LNoeMclnm 43063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-sca 17313  df-vsca 17314  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-subg 19153  df-ghm 19243  df-cntz 19347  df-lsm 19668  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-lmod 20876  df-lss 20947  df-lsp 20987  df-lmhm 21038  df-lfig 43056  df-lnm 43064
This theorem is referenced by:  pwslnmlem2  43081
  Copyright terms: Public domain W3C validator