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Theorem pj1lmhm 18921
 Description: The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
pj1lmhm.l 𝐿 = (LSubSp‘𝑊)
pj1lmhm.s = (LSSum‘𝑊)
pj1lmhm.z 0 = (0g𝑊)
pj1lmhm.p 𝑃 = (proj1𝑊)
pj1lmhm.1 (𝜑𝑊 ∈ LMod)
pj1lmhm.2 (𝜑𝑇𝐿)
pj1lmhm.3 (𝜑𝑈𝐿)
pj1lmhm.4 (𝜑 → (𝑇𝑈) = { 0 })
Assertion
Ref Expression
pj1lmhm (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊s (𝑇 𝑈)) LMHom 𝑊))

Proof of Theorem pj1lmhm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . 3 (+g𝑊) = (+g𝑊)
2 pj1lmhm.s . . 3 = (LSSum‘𝑊)
3 pj1lmhm.z . . 3 0 = (0g𝑊)
4 eqid 2610 . . 3 (Cntz‘𝑊) = (Cntz‘𝑊)
5 pj1lmhm.1 . . . . 5 (𝜑𝑊 ∈ LMod)
6 pj1lmhm.l . . . . . 6 𝐿 = (LSubSp‘𝑊)
76lsssssubg 18779 . . . . 5 (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊))
85, 7syl 17 . . . 4 (𝜑𝐿 ⊆ (SubGrp‘𝑊))
9 pj1lmhm.2 . . . 4 (𝜑𝑇𝐿)
108, 9sseldd 3569 . . 3 (𝜑𝑇 ∈ (SubGrp‘𝑊))
11 pj1lmhm.3 . . . 4 (𝜑𝑈𝐿)
128, 11sseldd 3569 . . 3 (𝜑𝑈 ∈ (SubGrp‘𝑊))
13 pj1lmhm.4 . . 3 (𝜑 → (𝑇𝑈) = { 0 })
14 lmodabl 18733 . . . . 5 (𝑊 ∈ LMod → 𝑊 ∈ Abel)
155, 14syl 17 . . . 4 (𝜑𝑊 ∈ Abel)
164, 15, 10, 12ablcntzd 18083 . . 3 (𝜑𝑇 ⊆ ((Cntz‘𝑊)‘𝑈))
17 pj1lmhm.p . . 3 𝑃 = (proj1𝑊)
181, 2, 3, 4, 10, 12, 13, 16, 17pj1ghm 17939 . 2 (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊s (𝑇 𝑈)) GrpHom 𝑊))
19 eqid 2610 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2019a1i 11 . 2 (𝜑 → (Scalar‘𝑊) = (Scalar‘𝑊))
211, 2, 3, 4, 10, 12, 13, 16, 17pj1id 17935 . . . . . . . . 9 ((𝜑𝑦 ∈ (𝑇 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦)(+g𝑊)((𝑈𝑃𝑇)‘𝑦)))
2221adantrl 748 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦)(+g𝑊)((𝑈𝑃𝑇)‘𝑦)))
2322oveq2d 6565 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥( ·𝑠𝑊)𝑦) = (𝑥( ·𝑠𝑊)(((𝑇𝑃𝑈)‘𝑦)(+g𝑊)((𝑈𝑃𝑇)‘𝑦))))
245adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑊 ∈ LMod)
25 simprl 790 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
269adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇𝐿)
27 eqid 2610 . . . . . . . . . . 11 (Base‘𝑊) = (Base‘𝑊)
2827, 6lssss 18758 . . . . . . . . . 10 (𝑇𝐿𝑇 ⊆ (Base‘𝑊))
2926, 28syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ⊆ (Base‘𝑊))
3010adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ∈ (SubGrp‘𝑊))
3112adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑈 ∈ (SubGrp‘𝑊))
3213adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑇𝑈) = { 0 })
3316adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ⊆ ((Cntz‘𝑊)‘𝑈))
341, 2, 3, 4, 30, 31, 32, 33, 17pj1f 17933 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
35 simprr 792 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑦 ∈ (𝑇 𝑈))
3634, 35ffvelrnd 6268 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
3729, 36sseldd 3569 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝑊))
3811adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑈𝐿)
3927, 6lssss 18758 . . . . . . . . . 10 (𝑈𝐿𝑈 ⊆ (Base‘𝑊))
4038, 39syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑈 ⊆ (Base‘𝑊))
411, 2, 3, 4, 30, 31, 32, 33, 17pj2f 17934 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈)
4241, 35ffvelrnd 6268 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
4340, 42sseldd 3569 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝑊))
44 eqid 2610 . . . . . . . . 9 ( ·𝑠𝑊) = ( ·𝑠𝑊)
45 eqid 2610 . . . . . . . . 9 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4627, 1, 19, 44, 45lmodvsdi 18709 . . . . . . . 8 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝑊) ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝑊))) → (𝑥( ·𝑠𝑊)(((𝑇𝑃𝑈)‘𝑦)(+g𝑊)((𝑈𝑃𝑇)‘𝑦))) = ((𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦))(+g𝑊)(𝑥( ·𝑠𝑊)((𝑈𝑃𝑇)‘𝑦))))
4724, 25, 37, 43, 46syl13anc 1320 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥( ·𝑠𝑊)(((𝑇𝑃𝑈)‘𝑦)(+g𝑊)((𝑈𝑃𝑇)‘𝑦))) = ((𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦))(+g𝑊)(𝑥( ·𝑠𝑊)((𝑈𝑃𝑇)‘𝑦))))
4823, 47eqtrd 2644 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥( ·𝑠𝑊)𝑦) = ((𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦))(+g𝑊)(𝑥( ·𝑠𝑊)((𝑈𝑃𝑇)‘𝑦))))
496, 2lsmcl 18904 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝑇𝐿𝑈𝐿) → (𝑇 𝑈) ∈ 𝐿)
505, 9, 11, 49syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝑇 𝑈) ∈ 𝐿)
5150adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑇 𝑈) ∈ 𝐿)
5219, 44, 45, 6lssvscl 18776 . . . . . . . 8 (((𝑊 ∈ LMod ∧ (𝑇 𝑈) ∈ 𝐿) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥( ·𝑠𝑊)𝑦) ∈ (𝑇 𝑈))
5324, 51, 25, 35, 52syl22anc 1319 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥( ·𝑠𝑊)𝑦) ∈ (𝑇 𝑈))
5419, 44, 45, 6lssvscl 18776 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑇𝐿) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)) → (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
5524, 26, 25, 36, 54syl22anc 1319 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
5619, 44, 45, 6lssvscl 18776 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝑈𝐿) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)) → (𝑥( ·𝑠𝑊)((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
5724, 38, 25, 42, 56syl22anc 1319 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥( ·𝑠𝑊)((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
581, 2, 3, 4, 30, 31, 32, 33, 17, 53, 55, 57pj1eq 17936 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑥( ·𝑠𝑊)𝑦) = ((𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦))(+g𝑊)(𝑥( ·𝑠𝑊)((𝑈𝑃𝑇)‘𝑦))) ↔ (((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑈𝑃𝑇)‘𝑦)))))
5948, 58mpbid 221 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑈𝑃𝑇)‘𝑦))))
6059simpld 474 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)))
6160ralrimivva 2954 . . 3 (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (𝑇 𝑈)((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)))
628, 50sseldd 3569 . . . . . 6 (𝜑 → (𝑇 𝑈) ∈ (SubGrp‘𝑊))
63 eqid 2610 . . . . . . 7 (𝑊s (𝑇 𝑈)) = (𝑊s (𝑇 𝑈))
6463subgbas 17421 . . . . . 6 ((𝑇 𝑈) ∈ (SubGrp‘𝑊) → (𝑇 𝑈) = (Base‘(𝑊s (𝑇 𝑈))))
6562, 64syl 17 . . . . 5 (𝜑 → (𝑇 𝑈) = (Base‘(𝑊s (𝑇 𝑈))))
6665raleqdv 3121 . . . 4 (𝜑 → (∀𝑦 ∈ (𝑇 𝑈)((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)) ↔ ∀𝑦 ∈ (Base‘(𝑊s (𝑇 𝑈)))((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦))))
6766ralbidv 2969 . . 3 (𝜑 → (∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (𝑇 𝑈)((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)) ↔ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘(𝑊s (𝑇 𝑈)))((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦))))
6861, 67mpbid 221 . 2 (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘(𝑊s (𝑇 𝑈)))((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)))
6963, 6lsslmod 18781 . . . 4 ((𝑊 ∈ LMod ∧ (𝑇 𝑈) ∈ 𝐿) → (𝑊s (𝑇 𝑈)) ∈ LMod)
705, 50, 69syl2anc 691 . . 3 (𝜑 → (𝑊s (𝑇 𝑈)) ∈ LMod)
71 ovex 6577 . . . . 5 (𝑇 𝑈) ∈ V
7263, 19resssca 15854 . . . . 5 ((𝑇 𝑈) ∈ V → (Scalar‘𝑊) = (Scalar‘(𝑊s (𝑇 𝑈))))
7371, 72ax-mp 5 . . . 4 (Scalar‘𝑊) = (Scalar‘(𝑊s (𝑇 𝑈)))
74 eqid 2610 . . . 4 (Base‘(𝑊s (𝑇 𝑈))) = (Base‘(𝑊s (𝑇 𝑈)))
7563, 44ressvsca 15855 . . . . 5 ((𝑇 𝑈) ∈ V → ( ·𝑠𝑊) = ( ·𝑠 ‘(𝑊s (𝑇 𝑈))))
7671, 75ax-mp 5 . . . 4 ( ·𝑠𝑊) = ( ·𝑠 ‘(𝑊s (𝑇 𝑈)))
7773, 19, 45, 74, 76, 44islmhm3 18849 . . 3 (((𝑊s (𝑇 𝑈)) ∈ LMod ∧ 𝑊 ∈ LMod) → ((𝑇𝑃𝑈) ∈ ((𝑊s (𝑇 𝑈)) LMHom 𝑊) ↔ ((𝑇𝑃𝑈) ∈ ((𝑊s (𝑇 𝑈)) GrpHom 𝑊) ∧ (Scalar‘𝑊) = (Scalar‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘(𝑊s (𝑇 𝑈)))((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)))))
7870, 5, 77syl2anc 691 . 2 (𝜑 → ((𝑇𝑃𝑈) ∈ ((𝑊s (𝑇 𝑈)) LMHom 𝑊) ↔ ((𝑇𝑃𝑈) ∈ ((𝑊s (𝑇 𝑈)) GrpHom 𝑊) ∧ (Scalar‘𝑊) = (Scalar‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘(𝑊s (𝑇 𝑈)))((𝑇𝑃𝑈)‘(𝑥( ·𝑠𝑊)𝑦)) = (𝑥( ·𝑠𝑊)((𝑇𝑃𝑈)‘𝑦)))))
7918, 20, 68, 78mpbir3and 1238 1 (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊s (𝑇 𝑈)) LMHom 𝑊))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  SubGrpcsubg 17411   GrpHom cghm 17480  Cntzccntz 17571  LSSumclsm 17872  proj1cpj1 17873  Abelcabl 18017  LModclmod 18686  LSubSpclss 18753   LMHom clmhm 18840 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-sca 15784  df-vsca 15785  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-ghm 17481  df-cntz 17573  df-lsm 17874  df-pj1 17875  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-lss 18754  df-lmhm 18843 This theorem is referenced by:  pj1lmhm2  18922  pjff  19875
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