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Theorem dprdres 18619
Description: Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdres.1 (𝜑𝐺dom DProd 𝑆)
dprdres.2 (𝜑 → dom 𝑆 = 𝐼)
dprdres.3 (𝜑𝐴𝐼)
Assertion
Ref Expression
dprdres (𝜑 → (𝐺dom DProd (𝑆𝐴) ∧ (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆)))

Proof of Theorem dprdres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdres.1 . . . 4 (𝜑𝐺dom DProd 𝑆)
2 dprdgrp 18596 . . . 4 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
31, 2syl 17 . . 3 (𝜑𝐺 ∈ Grp)
4 dprdres.2 . . . . 5 (𝜑 → dom 𝑆 = 𝐼)
51, 4dprdf2 18598 . . . 4 (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
6 dprdres.3 . . . 4 (𝜑𝐴𝐼)
75, 6fssresd 6224 . . 3 (𝜑 → (𝑆𝐴):𝐴⟶(SubGrp‘𝐺))
81ad2antrr 764 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐺dom DProd 𝑆)
94ad2antrr 764 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → dom 𝑆 = 𝐼)
106ad2antrr 764 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐴𝐼)
11 simplr 809 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝐴)
1210, 11sseldd 3737 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝐼)
13 eldifi 3867 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦𝐴)
1413adantl 473 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝐴)
1510, 14sseldd 3737 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝐼)
16 eldifsni 4458 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦𝑥)
1716adantl 473 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝑥)
1817necomd 2979 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝑦)
19 eqid 2752 . . . . . . . 8 (Cntz‘𝐺) = (Cntz‘𝐺)
208, 9, 12, 15, 18, 19dprdcntz 18599 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
21 fvres 6360 . . . . . . . 8 (𝑥𝐴 → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2211, 21syl 17 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
23 fvres 6360 . . . . . . . . 9 (𝑦𝐴 → ((𝑆𝐴)‘𝑦) = (𝑆𝑦))
2414, 23syl 17 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑦) = (𝑆𝑦))
2524fveq2d 6348 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆𝑦)))
2620, 22, 253sstr4d 3781 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)))
2726ralrimiva 3096 . . . . 5 ((𝜑𝑥𝐴) → ∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)))
2821adantl 473 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2928ineq1d 3948 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
30 eqid 2752 . . . . . . . . . . . . 13 (Base‘𝐺) = (Base‘𝐺)
3130subgacs 17822 . . . . . . . . . . . 12 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
32 acsmre 16506 . . . . . . . . . . . 12 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
333, 31, 323syl 18 . . . . . . . . . . 11 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
3433adantr 472 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
35 eqid 2752 . . . . . . . . . 10 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
36 resss 5572 . . . . . . . . . . . . 13 (𝑆𝐴) ⊆ 𝑆
37 imass1 5650 . . . . . . . . . . . . 13 ((𝑆𝐴) ⊆ 𝑆 → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥})))
3836, 37ax-mp 5 . . . . . . . . . . . 12 ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥}))
396adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐴𝐼)
40 ssdif 3880 . . . . . . . . . . . . 13 (𝐴𝐼 → (𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}))
41 imass2 5651 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
4239, 40, 413syl 18 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
4338, 42syl5ss 3747 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
4443unissd 4606 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
45 imassrn 5627 . . . . . . . . . . . 12 (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
46 frn 6206 . . . . . . . . . . . . . . 15 (𝑆:𝐼⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
475, 46syl 17 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
4830subgss 17788 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
49 selpw 4301 . . . . . . . . . . . . . . . 16 (𝑥 ∈ 𝒫 (Base‘𝐺) ↔ 𝑥 ⊆ (Base‘𝐺))
5048, 49sylibr 224 . . . . . . . . . . . . . . 15 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ∈ 𝒫 (Base‘𝐺))
5150ssriv 3740 . . . . . . . . . . . . . 14 (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)
5247, 51syl6ss 3748 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5352adantr 472 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5445, 53syl5ss 3747 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
55 sspwuni 4755 . . . . . . . . . . 11 ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
5654, 55sylib 208 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
5734, 35, 44, 56mrcssd 16478 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
58 sslin 3974 . . . . . . . . 9 (((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))))
5957, 58syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))))
601adantr 472 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐺dom DProd 𝑆)
614adantr 472 . . . . . . . . 9 ((𝜑𝑥𝐴) → dom 𝑆 = 𝐼)
626sselda 3736 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐼)
63 eqid 2752 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
6460, 61, 62, 63, 35dprddisj 18600 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)})
6559, 64sseqtrd 3774 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ {(0g𝐺)})
665ffvelrnda 6514 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
6762, 66syldan 488 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
6863subg0cl 17795 . . . . . . . . . 10 ((𝑆𝑥) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝑆𝑥))
6967, 68syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝑆𝑥))
7044, 56sstrd 3746 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺))
7135mrccl 16465 . . . . . . . . . . 11 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7234, 70, 71syl2anc 696 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7363subg0cl 17795 . . . . . . . . . 10 (((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))))
7472, 73syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))))
7569, 74elind 3933 . . . . . . . 8 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
7675snssd 4477 . . . . . . 7 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
7765, 76eqssd 3753 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)})
7829, 77eqtrd 2786 . . . . 5 ((𝜑𝑥𝐴) → (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)})
7927, 78jca 555 . . . 4 ((𝜑𝑥𝐴) → (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))
8079ralrimiva 3096 . . 3 (𝜑 → ∀𝑥𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))
811, 4dprddomcld 18592 . . . . 5 (𝜑𝐼 ∈ V)
8281, 6ssexd 4949 . . . 4 (𝜑𝐴 ∈ V)
83 fdm 6204 . . . . 5 ((𝑆𝐴):𝐴⟶(SubGrp‘𝐺) → dom (𝑆𝐴) = 𝐴)
847, 83syl 17 . . . 4 (𝜑 → dom (𝑆𝐴) = 𝐴)
8519, 63, 35dmdprd 18589 . . . 4 ((𝐴 ∈ V ∧ dom (𝑆𝐴) = 𝐴) → (𝐺dom DProd (𝑆𝐴) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐴):𝐴⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))))
8682, 84, 85syl2anc 696 . . 3 (𝜑 → (𝐺dom DProd (𝑆𝐴) ↔ (𝐺 ∈ Grp ∧ (𝑆𝐴):𝐴⟶(SubGrp‘𝐺) ∧ ∀𝑥𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))))
873, 7, 80, 86mpbir3and 1425 . 2 (𝜑𝐺dom DProd (𝑆𝐴))
88 rnss 5501 . . . . . 6 ((𝑆𝐴) ⊆ 𝑆 → ran (𝑆𝐴) ⊆ ran 𝑆)
89 uniss 4602 . . . . . 6 (ran (𝑆𝐴) ⊆ ran 𝑆 ran (𝑆𝐴) ⊆ ran 𝑆)
9036, 88, 89mp2b 10 . . . . 5 ran (𝑆𝐴) ⊆ ran 𝑆
9190a1i 11 . . . 4 (𝜑 ran (𝑆𝐴) ⊆ ran 𝑆)
92 sspwuni 4755 . . . . 5 (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ran 𝑆 ⊆ (Base‘𝐺))
9352, 92sylib 208 . . . 4 (𝜑 ran 𝑆 ⊆ (Base‘𝐺))
9433, 35, 91, 93mrcssd 16478 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran (𝑆𝐴)) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
9535dprdspan 18618 . . . 4 (𝐺dom DProd (𝑆𝐴) → (𝐺 DProd (𝑆𝐴)) = ((mrCls‘(SubGrp‘𝐺))‘ ran (𝑆𝐴)))
9687, 95syl 17 . . 3 (𝜑 → (𝐺 DProd (𝑆𝐴)) = ((mrCls‘(SubGrp‘𝐺))‘ ran (𝑆𝐴)))
9735dprdspan 18618 . . . 4 (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
981, 97syl 17 . . 3 (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
9994, 96, 983sstr4d 3781 . 2 (𝜑 → (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆))
10087, 99jca 555 1 (𝜑 → (𝐺dom DProd (𝑆𝐴) ∧ (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wral 3042  Vcvv 3332  cdif 3704  cin 3706  wss 3707  𝒫 cpw 4294  {csn 4313   cuni 4580   class class class wbr 4796  dom cdm 5258  ran crn 5259  cres 5260  cima 5261  wf 6037  cfv 6041  (class class class)co 6805  Basecbs 16051  0gc0g 16294  Moorecmre 16436  mrClscmrc 16437  ACScacs 16439  Grpcgrp 17615  SubGrpcsubg 17781  Cntzccntz 17940   DProd cdprd 18584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-om 7223  df-1st 7325  df-2nd 7326  df-supp 7456  df-tpos 7513  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fsupp 8433  df-oi 8572  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-n0 11477  df-z 11562  df-uz 11872  df-fz 12512  df-fzo 12652  df-seq 12988  df-hash 13304  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-0g 16296  df-gsum 16297  df-mre 16440  df-mrc 16441  df-acs 16443  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-mhm 17528  df-submnd 17529  df-grp 17618  df-minusg 17619  df-sbg 17620  df-mulg 17734  df-subg 17784  df-ghm 17851  df-gim 17894  df-cntz 17942  df-oppg 17968  df-cmn 18387  df-dprd 18586
This theorem is referenced by:  dprdf1  18624  dprdcntz2  18629  dprddisj2  18630  dprd2dlem1  18632  dprd2da  18633  dmdprdsplit  18638  dprdsplit  18639  dpjf  18648  dpjidcl  18649  dpjlid  18652  dpjghm  18654  ablfac1eulem  18663  ablfac1eu  18664
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