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Theorem dprdres 18473
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 18450 . . . 4 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
31, 2syl 17 . . 3 (𝜑𝐺 ∈ Grp)
4 dprdres.2 . . . . 5 (𝜑 → dom 𝑆 = 𝐼)
51, 4dprdf2 18452 . . . 4 (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
6 dprdres.3 . . . 4 (𝜑𝐴𝐼)
75, 6fssresd 6109 . . 3 (𝜑 → (𝑆𝐴):𝐴⟶(SubGrp‘𝐺))
81ad2antrr 762 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐺dom DProd 𝑆)
94ad2antrr 762 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → dom 𝑆 = 𝐼)
106ad2antrr 762 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐴𝐼)
11 simplr 807 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝐴)
1210, 11sseldd 3637 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝐼)
13 eldifi 3765 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦𝐴)
1413adantl 481 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝐴)
1510, 14sseldd 3637 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝐼)
16 eldifsni 4353 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦𝑥)
1716adantl 481 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝑥)
1817necomd 2878 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝑦)
19 eqid 2651 . . . . . . . 8 (Cntz‘𝐺) = (Cntz‘𝐺)
208, 9, 12, 15, 18, 19dprdcntz 18453 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
21 fvres 6245 . . . . . . . 8 (𝑥𝐴 → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2211, 21syl 17 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
23 fvres 6245 . . . . . . . . 9 (𝑦𝐴 → ((𝑆𝐴)‘𝑦) = (𝑆𝑦))
2414, 23syl 17 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑦) = (𝑆𝑦))
2524fveq2d 6233 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆𝑦)))
2620, 22, 253sstr4d 3681 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)))
2726ralrimiva 2995 . . . . 5 ((𝜑𝑥𝐴) → ∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)))
2821adantl 481 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2928ineq1d 3846 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
30 eqid 2651 . . . . . . . . . . . . 13 (Base‘𝐺) = (Base‘𝐺)
3130subgacs 17676 . . . . . . . . . . . 12 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
32 acsmre 16360 . . . . . . . . . . . 12 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
333, 31, 323syl 18 . . . . . . . . . . 11 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
3433adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
35 eqid 2651 . . . . . . . . . 10 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
36 resss 5457 . . . . . . . . . . . . 13 (𝑆𝐴) ⊆ 𝑆
37 imass1 5535 . . . . . . . . . . . . 13 ((𝑆𝐴) ⊆ 𝑆 → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥})))
3836, 37ax-mp 5 . . . . . . . . . . . 12 ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥}))
396adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐴𝐼)
40 ssdif 3778 . . . . . . . . . . . . 13 (𝐴𝐼 → (𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}))
41 imass2 5536 . . . . . . . . . . . . 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 3647 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
4443unissd 4494 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
45 imassrn 5512 . . . . . . . . . . . 12 (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
46 frn 6091 . . . . . . . . . . . . . . 15 (𝑆:𝐼⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
475, 46syl 17 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
4830subgss 17642 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
49 selpw 4198 . . . . . . . . . . . . . . . 16 (𝑥 ∈ 𝒫 (Base‘𝐺) ↔ 𝑥 ⊆ (Base‘𝐺))
5048, 49sylibr 224 . . . . . . . . . . . . . . 15 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ∈ 𝒫 (Base‘𝐺))
5150ssriv 3640 . . . . . . . . . . . . . 14 (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)
5247, 51syl6ss 3648 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5352adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5445, 53syl5ss 3647 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
55 sspwuni 4643 . . . . . . . . . . 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 16331 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
58 sslin 3872 . . . . . . . . 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 480 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐺dom DProd 𝑆)
614adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐴) → dom 𝑆 = 𝐼)
626sselda 3636 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐼)
63 eqid 2651 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
6460, 61, 62, 63, 35dprddisj 18454 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)})
6559, 64sseqtrd 3674 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ {(0g𝐺)})
665ffvelrnda 6399 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
6762, 66syldan 486 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
6863subg0cl 17649 . . . . . . . . . 10 ((𝑆𝑥) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝑆𝑥))
6967, 68syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝑆𝑥))
7044, 56sstrd 3646 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺))
7135mrccl 16318 . . . . . . . . . . 11 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7234, 70, 71syl2anc 694 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7363subg0cl 17649 . . . . . . . . . 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 3831 . . . . . . . 8 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
7675snssd 4372 . . . . . . 7 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
7765, 76eqssd 3653 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)})
7829, 77eqtrd 2685 . . . . 5 ((𝜑𝑥𝐴) → (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)})
7927, 78jca 553 . . . 4 ((𝜑𝑥𝐴) → (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))
8079ralrimiva 2995 . . 3 (𝜑 → ∀𝑥𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))
811, 4dprddomcld 18446 . . . . 5 (𝜑𝐼 ∈ V)
8281, 6ssexd 4838 . . . 4 (𝜑𝐴 ∈ V)
83 fdm 6089 . . . . 5 ((𝑆𝐴):𝐴⟶(SubGrp‘𝐺) → dom (𝑆𝐴) = 𝐴)
847, 83syl 17 . . . 4 (𝜑 → dom (𝑆𝐴) = 𝐴)
8519, 63, 35dmdprd 18443 . . . 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 694 . . 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 1264 . 2 (𝜑𝐺dom DProd (𝑆𝐴))
88 rnss 5386 . . . . . 6 ((𝑆𝐴) ⊆ 𝑆 → ran (𝑆𝐴) ⊆ ran 𝑆)
89 uniss 4490 . . . . . 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 4643 . . . . 5 (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ran 𝑆 ⊆ (Base‘𝐺))
9352, 92sylib 208 . . . 4 (𝜑 ran 𝑆 ⊆ (Base‘𝐺))
9433, 35, 91, 93mrcssd 16331 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran (𝑆𝐴)) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
9535dprdspan 18472 . . . 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 18472 . . . 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 3681 . 2 (𝜑 → (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆))
10087, 99jca 553 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 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cdif 3604  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210   cuni 4468   class class class wbr 4685  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  0gc0g 16147  Moorecmre 16289  mrClscmrc 16290  ACScacs 16292  Grpcgrp 17469  SubGrpcsubg 17635  Cntzccntz 17794   DProd cdprd 18438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-gim 17748  df-cntz 17796  df-oppg 17822  df-cmn 18241  df-dprd 18440
This theorem is referenced by:  dprdf1  18478  dprdcntz2  18483  dprddisj2  18484  dprd2dlem1  18486  dprd2da  18487  dmdprdsplit  18492  dprdsplit  18493  dpjf  18502  dpjidcl  18503  dpjlid  18506  dpjghm  18508  ablfac1eulem  18517  ablfac1eu  18518
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