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 42284
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 20870 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
213ad2ant1 1130 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LMod)
3 eqid 2724 . . . . . 6 (LSubSp‘𝑆) = (LSubSp‘𝑆)
4 eqid 2724 . . . . . 6 (𝑆s 𝑎) = (𝑆s 𝑎)
53, 4reslmhm 20889 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇))
653ad2antl1 1182 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇))
7 cnvresima 6219 . . . . . . . 8 ((𝐹𝑎) “ { 0 }) = ((𝐹 “ { 0 }) ∩ 𝑎)
8 lmhmfgsplit.k . . . . . . . . . 10 𝐾 = (𝐹 “ { 0 })
98eqcomi 2733 . . . . . . . . 9 (𝐹 “ { 0 }) = 𝐾
109ineq1i 4200 . . . . . . . 8 ((𝐹 “ { 0 }) ∩ 𝑎) = (𝐾𝑎)
11 incom 4193 . . . . . . . 8 (𝐾𝑎) = (𝑎𝐾)
127, 10, 113eqtri 2756 . . . . . . 7 ((𝐹𝑎) “ { 0 }) = (𝑎𝐾)
1312oveq2i 7412 . . . . . 6 ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = ((𝑆s 𝑎) ↾s (𝑎𝐾))
14 vex 3470 . . . . . . . 8 𝑎 ∈ V
15 inss1 4220 . . . . . . . 8 (𝑎𝐾) ⊆ 𝑎
16 ressabs 17192 . . . . . . . 8 ((𝑎 ∈ V ∧ (𝑎𝐾) ⊆ 𝑎) → ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
1714, 15, 16mp2an 689 . . . . . . 7 ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾))
18 lmhmfgsplit.u . . . . . . . . 9 𝑈 = (𝑆s 𝐾)
1918oveq1i 7411 . . . . . . . 8 (𝑈s (𝑎𝐾)) = ((𝑆s 𝐾) ↾s (𝑎𝐾))
20 simpl1 1188 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
21 cnvexg 7908 . . . . . . . . . . . 12 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ V)
22 imaexg 7899 . . . . . . . . . . . 12 (𝐹 ∈ V → (𝐹 “ { 0 }) ∈ V)
2321, 22syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 “ { 0 }) ∈ V)
248, 23eqeltrid 2829 . . . . . . . . . 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 4221 . . . . . . . . 9 (𝑎𝐾) ⊆ 𝐾
27 ressabs 17192 . . . . . . . . 9 ((𝐾 ∈ V ∧ (𝑎𝐾) ⊆ 𝐾) → ((𝑆s 𝐾) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
2825, 26, 27sylancl 585 . . . . . . . 8 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝐾) ↾s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
2919, 28eqtrid 2776 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈s (𝑎𝐾)) = (𝑆s (𝑎𝐾)))
3017, 29eqtr4id 2783 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s (𝑎𝐾)) = (𝑈s (𝑎𝐾)))
3113, 30eqtrid 2776 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = (𝑈s (𝑎𝐾)))
32 simpl2 1189 . . . . . 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 20888 . . . . . . . . 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 20801 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (LSubSp‘𝑆) ∧ 𝐾 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ∈ (LSubSp‘𝑆))
3933, 34, 37, 38syl3anc 1368 . . . . . . 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 2724 . . . . . . . . 9 (LSubSp‘𝑈) = (LSubSp‘𝑈)
4218, 3, 41lsslss 20797 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝑆)) → ((𝑎𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎𝐾) ⊆ 𝐾)))
4333, 37, 42syl2anc 583 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑎𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎𝐾) ⊆ 𝐾)))
4439, 40, 43mpbir2and 710 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎𝐾) ∈ (LSubSp‘𝑈))
45 eqid 2724 . . . . . . 7 (𝑈s (𝑎𝐾)) = (𝑈s (𝑎𝐾))
4641, 45lnmlssfg 42277 . . . . . 6 ((𝑈 ∈ LNoeM ∧ (𝑎𝐾) ∈ (LSubSp‘𝑈)) → (𝑈s (𝑎𝐾)) ∈ LFinGen)
4732, 44, 46syl2anc 583 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈s (𝑎𝐾)) ∈ LFinGen)
4831, 47eqeltrd 2825 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) ∈ LFinGen)
49 incom 4193 . . . . . . . . 9 (ran 𝐹 ∩ ran (𝐹𝑎)) = (ran (𝐹𝑎) ∩ ran 𝐹)
50 resss 5996 . . . . . . . . . . 11 (𝐹𝑎) ⊆ 𝐹
51 rnss 5928 . . . . . . . . . . 11 ((𝐹𝑎) ⊆ 𝐹 → ran (𝐹𝑎) ⊆ ran 𝐹)
5250, 51ax-mp 5 . . . . . . . . . 10 ran (𝐹𝑎) ⊆ ran 𝐹
53 df-ss 3957 . . . . . . . . . 10 (ran (𝐹𝑎) ⊆ ran 𝐹 ↔ (ran (𝐹𝑎) ∩ ran 𝐹) = ran (𝐹𝑎))
5452, 53mpbi 229 . . . . . . . . 9 (ran (𝐹𝑎) ∩ ran 𝐹) = ran (𝐹𝑎)
5549, 54eqtr2i 2753 . . . . . . . 8 ran (𝐹𝑎) = (ran 𝐹 ∩ ran (𝐹𝑎))
5655oveq2i 7412 . . . . . . 7 (𝑇s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎)))
57 lmhmfgsplit.v . . . . . . . . 9 𝑉 = (𝑇s ran 𝐹)
5857oveq1i 7411 . . . . . . . 8 (𝑉s ran (𝐹𝑎)) = ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎))
59 rnexg 7888 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ V)
60 resexg 6017 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹𝑎) ∈ V)
61 rnexg 7888 . . . . . . . . . 10 ((𝐹𝑎) ∈ V → ran (𝐹𝑎) ∈ V)
6260, 61syl 17 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran (𝐹𝑎) ∈ V)
63 ressress 17191 . . . . . . . . 9 ((ran 𝐹 ∈ V ∧ ran (𝐹𝑎) ∈ V) → ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
6459, 62, 63syl2anc 583 . . . . . . . 8 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑇s ran 𝐹) ↾s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
6558, 64eqtrid 2776 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑉s ran (𝐹𝑎)) = (𝑇s (ran 𝐹 ∩ ran (𝐹𝑎))))
6656, 65eqtr4id 2783 . . . . . 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 1190 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑉 ∈ LNoeM)
69 lmhmrnlss 20887 . . . . . . . 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 20869 . . . . . . . . 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 20887 . . . . . . . . 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 2724 . . . . . . . . 9 (LSubSp‘𝑇) = (LSubSp‘𝑇)
77 eqid 2724 . . . . . . . . 9 (LSubSp‘𝑉) = (LSubSp‘𝑉)
7857, 76, 77lsslss 20797 . . . . . . . 8 ((𝑇 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑇)) → (ran (𝐹𝑎) ∈ (LSubSp‘𝑉) ↔ (ran (𝐹𝑎) ∈ (LSubSp‘𝑇) ∧ ran (𝐹𝑎) ⊆ ran 𝐹)))
7973, 75, 78syl2anc 583 . . . . . . 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 710 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹𝑎) ∈ (LSubSp‘𝑉))
81 eqid 2724 . . . . . . 7 (𝑉s ran (𝐹𝑎)) = (𝑉s ran (𝐹𝑎))
8277, 81lnmlssfg 42277 . . . . . 6 ((𝑉 ∈ LNoeM ∧ ran (𝐹𝑎) ∈ (LSubSp‘𝑉)) → (𝑉s ran (𝐹𝑎)) ∈ LFinGen)
8368, 80, 82syl2anc 583 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑉s ran (𝐹𝑎)) ∈ LFinGen)
8467, 83eqeltrd 2825 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇s ran (𝐹𝑎)) ∈ LFinGen)
85 eqid 2724 . . . . 5 ((𝐹𝑎) “ { 0 }) = ((𝐹𝑎) “ { 0 })
86 eqid 2724 . . . . 5 ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) = ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 }))
87 eqid 2724 . . . . 5 (𝑇s ran (𝐹𝑎)) = (𝑇s ran (𝐹𝑎))
8835, 85, 86, 87lmhmfgsplit 42283 . . . 4 (((𝐹𝑎) ∈ ((𝑆s 𝑎) LMHom 𝑇) ∧ ((𝑆s 𝑎) ↾s ((𝐹𝑎) “ { 0 })) ∈ LFinGen ∧ (𝑇s ran (𝐹𝑎)) ∈ LFinGen) → (𝑆s 𝑎) ∈ LFinGen)
896, 48, 84, 88syl3anc 1368 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑆s 𝑎) ∈ LFinGen)
9089ralrimiva 3138 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆s 𝑎) ∈ LFinGen)
913islnm 42274 . 2 (𝑆 ∈ LNoeM ↔ (𝑆 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆s 𝑎) ∈ LFinGen))
922, 90, 91sylanbrc 582 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  Vcvv 3466  cin 3939  wss 3940  {csn 4620  ccnv 5665  ran crn 5667  cres 5668  cima 5669  cfv 6533  (class class class)co 7401  s cress 17171  0gc0g 17383  LModclmod 20695  LSubSpclss 20767   LMHom clmhm 20856  LFinGenclfig 42264  LNoeMclnm 42272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17143  df-ress 17172  df-plusg 17208  df-sca 17211  df-vsca 17212  df-0g 17385  df-mgm 18562  df-sgrp 18641  df-mnd 18657  df-submnd 18703  df-grp 18855  df-minusg 18856  df-sbg 18857  df-subg 19039  df-ghm 19128  df-cntz 19222  df-lsm 19545  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-lmod 20697  df-lss 20768  df-lsp 20808  df-lmhm 20859  df-lfig 42265  df-lnm 42273
This theorem is referenced by:  pwslnmlem2  42290
  Copyright terms: Public domain W3C validator