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Theorem dprdres 18348
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 18325 . . . 4 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
31, 2syl 17 . . 3 (𝜑𝐺 ∈ Grp)
4 dprdres.2 . . . . 5 (𝜑 → dom 𝑆 = 𝐼)
51, 4dprdf2 18327 . . . 4 (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
6 dprdres.3 . . . 4 (𝜑𝐴𝐼)
75, 6fssresd 6028 . . 3 (𝜑 → (𝑆𝐴):𝐴⟶(SubGrp‘𝐺))
81ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐺dom DProd 𝑆)
94ad2antrr 761 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → dom 𝑆 = 𝐼)
106ad2antrr 761 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐴𝐼)
11 simplr 791 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝐴)
1210, 11sseldd 3584 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝐼)
13 eldifi 3710 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦𝐴)
1413adantl 482 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝐴)
1510, 14sseldd 3584 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝐼)
16 eldifsni 4289 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦𝑥)
1716adantl 482 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝑥)
1817necomd 2845 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝑦)
19 eqid 2621 . . . . . . . 8 (Cntz‘𝐺) = (Cntz‘𝐺)
208, 9, 12, 15, 18, 19dprdcntz 18328 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
21 fvres 6164 . . . . . . . 8 (𝑥𝐴 → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2211, 21syl 17 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
23 fvres 6164 . . . . . . . . 9 (𝑦𝐴 → ((𝑆𝐴)‘𝑦) = (𝑆𝑦))
2414, 23syl 17 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑦) = (𝑆𝑦))
2524fveq2d 6152 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆𝑦)))
2620, 22, 253sstr4d 3627 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)))
2726ralrimiva 2960 . . . . 5 ((𝜑𝑥𝐴) → ∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)))
2821adantl 482 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2928ineq1d 3791 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
30 eqid 2621 . . . . . . . . . . . . 13 (Base‘𝐺) = (Base‘𝐺)
3130subgacs 17550 . . . . . . . . . . . 12 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
32 acsmre 16234 . . . . . . . . . . . 12 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
333, 31, 323syl 18 . . . . . . . . . . 11 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
3433adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
35 eqid 2621 . . . . . . . . . 10 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
36 resss 5381 . . . . . . . . . . . . 13 (𝑆𝐴) ⊆ 𝑆
37 imass1 5459 . . . . . . . . . . . . 13 ((𝑆𝐴) ⊆ 𝑆 → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥})))
3836, 37ax-mp 5 . . . . . . . . . . . 12 ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥}))
396adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐴𝐼)
40 ssdif 3723 . . . . . . . . . . . . 13 (𝐴𝐼 → (𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}))
41 imass2 5460 . . . . . . . . . . . . 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 3594 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
4443unissd 4428 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
45 imassrn 5436 . . . . . . . . . . . 12 (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
46 frn 6010 . . . . . . . . . . . . . . 15 (𝑆:𝐼⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
475, 46syl 17 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
4830subgss 17516 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
49 selpw 4137 . . . . . . . . . . . . . . . 16 (𝑥 ∈ 𝒫 (Base‘𝐺) ↔ 𝑥 ⊆ (Base‘𝐺))
5048, 49sylibr 224 . . . . . . . . . . . . . . 15 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ∈ 𝒫 (Base‘𝐺))
5150ssriv 3587 . . . . . . . . . . . . . 14 (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)
5247, 51syl6ss 3595 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5352adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5445, 53syl5ss 3594 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
55 sspwuni 4577 . . . . . . . . . . 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 16205 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
58 sslin 3817 . . . . . . . . 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 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐺dom DProd 𝑆)
614adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → dom 𝑆 = 𝐼)
626sselda 3583 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐼)
63 eqid 2621 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
6460, 61, 62, 63, 35dprddisj 18329 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)})
6559, 64sseqtrd 3620 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ {(0g𝐺)})
665ffvelrnda 6315 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
6762, 66syldan 487 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
6863subg0cl 17523 . . . . . . . . . 10 ((𝑆𝑥) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝑆𝑥))
6967, 68syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝑆𝑥))
7044, 56sstrd 3593 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺))
7135mrccl 16192 . . . . . . . . . . 11 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7234, 70, 71syl2anc 692 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7363subg0cl 17523 . . . . . . . . . 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 3776 . . . . . . . 8 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
7675snssd 4309 . . . . . . 7 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
7765, 76eqssd 3600 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)})
7829, 77eqtrd 2655 . . . . 5 ((𝜑𝑥𝐴) → (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)})
7927, 78jca 554 . . . 4 ((𝜑𝑥𝐴) → (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))
8079ralrimiva 2960 . . 3 (𝜑 → ∀𝑥𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))
811, 4dprddomcld 18321 . . . . 5 (𝜑𝐼 ∈ V)
8281, 6ssexd 4765 . . . 4 (𝜑𝐴 ∈ V)
83 fdm 6008 . . . . 5 ((𝑆𝐴):𝐴⟶(SubGrp‘𝐺) → dom (𝑆𝐴) = 𝐴)
847, 83syl 17 . . . 4 (𝜑 → dom (𝑆𝐴) = 𝐴)
8519, 63, 35dmdprd 18318 . . . 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 692 . . 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 1243 . 2 (𝜑𝐺dom DProd (𝑆𝐴))
88 rnss 5314 . . . . . 6 ((𝑆𝐴) ⊆ 𝑆 → ran (𝑆𝐴) ⊆ ran 𝑆)
89 uniss 4424 . . . . . 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 4577 . . . . 5 (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ran 𝑆 ⊆ (Base‘𝐺))
9352, 92sylib 208 . . . 4 (𝜑 ran 𝑆 ⊆ (Base‘𝐺))
9433, 35, 91, 93mrcssd 16205 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran (𝑆𝐴)) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
9535dprdspan 18347 . . . 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 18347 . . . 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 3627 . 2 (𝜑 → (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆))
10087, 99jca 554 1 (𝜑 → (𝐺dom DProd (𝑆𝐴) ∧ (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186  cdif 3552  cin 3554  wss 3555  𝒫 cpw 4130  {csn 4148   cuni 4402   class class class wbr 4613  dom cdm 5074  ran crn 5075  cres 5076  cima 5077  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  0gc0g 16021  Moorecmre 16163  mrClscmrc 16164  ACScacs 16166  Grpcgrp 17343  SubGrpcsubg 17509  Cntzccntz 17669   DProd cdprd 18313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-tpos 7297  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-gsum 16024  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-subg 17512  df-ghm 17579  df-gim 17622  df-cntz 17671  df-oppg 17697  df-cmn 18116  df-dprd 18315
This theorem is referenced by:  dprdf1  18353  dprdcntz2  18358  dprddisj2  18359  dprd2dlem1  18361  dprd2da  18362  dmdprdsplit  18367  dprdsplit  18368  dpjf  18377  dpjidcl  18378  dpjlid  18381  dpjghm  18383  ablfac1eulem  18392  ablfac1eu  18393
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