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Theorem dprdres 19143
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 19120 . . . 4 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
31, 2syl 17 . . 3 (𝜑𝐺 ∈ Grp)
4 dprdres.2 . . . . 5 (𝜑 → dom 𝑆 = 𝐼)
51, 4dprdf2 19122 . . . 4 (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
6 dprdres.3 . . . 4 (𝜑𝐴𝐼)
75, 6fssresd 6519 . . 3 (𝜑 → (𝑆𝐴):𝐴⟶(SubGrp‘𝐺))
81ad2antrr 725 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐺dom DProd 𝑆)
94ad2antrr 725 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → dom 𝑆 = 𝐼)
106ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝐴𝐼)
11 simplr 768 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝐴)
1210, 11sseldd 3916 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝐼)
13 eldifi 4054 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦𝐴)
1413adantl 485 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝐴)
1510, 14sseldd 3916 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝐼)
16 eldifsni 4683 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦𝑥)
1716adantl 485 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝑥)
1817necomd 3042 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝑦)
19 eqid 2798 . . . . . . . 8 (Cntz‘𝐺) = (Cntz‘𝐺)
208, 9, 12, 15, 18, 19dprdcntz 19123 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
2111fvresd 6665 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2214fvresd 6665 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑦) = (𝑆𝑦))
2322fveq2d 6649 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆𝑦)))
2420, 21, 233sstr4d 3962 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)))
2524ralrimiva 3149 . . . . 5 ((𝜑𝑥𝐴) → ∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)))
26 fvres 6664 . . . . . . . 8 (𝑥𝐴 → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2726adantl 485 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2827ineq1d 4138 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
29 eqid 2798 . . . . . . . . . . . . 13 (Base‘𝐺) = (Base‘𝐺)
3029subgacs 18305 . . . . . . . . . . . 12 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
31 acsmre 16915 . . . . . . . . . . . 12 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
323, 30, 313syl 18 . . . . . . . . . . 11 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
3332adantr 484 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
34 eqid 2798 . . . . . . . . . 10 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
35 resss 5843 . . . . . . . . . . . . 13 (𝑆𝐴) ⊆ 𝑆
36 imass1 5931 . . . . . . . . . . . . 13 ((𝑆𝐴) ⊆ 𝑆 → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥})))
3735, 36ax-mp 5 . . . . . . . . . . . 12 ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥}))
386adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐴𝐼)
39 ssdif 4067 . . . . . . . . . . . . 13 (𝐴𝐼 → (𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}))
40 imass2 5932 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
4138, 39, 403syl 18 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
4237, 41sstrid 3926 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
4342unissd 4810 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
44 imassrn 5907 . . . . . . . . . . . 12 (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
455frnd 6494 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
4629subgss 18272 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
47 velpw 4502 . . . . . . . . . . . . . . . 16 (𝑥 ∈ 𝒫 (Base‘𝐺) ↔ 𝑥 ⊆ (Base‘𝐺))
4846, 47sylibr 237 . . . . . . . . . . . . . . 15 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ∈ 𝒫 (Base‘𝐺))
4948ssriv 3919 . . . . . . . . . . . . . 14 (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)
5045, 49sstrdi 3927 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5150adantr 484 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5244, 51sstrid 3926 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
53 sspwuni 4985 . . . . . . . . . . 11 ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
5452, 53sylib 221 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
5533, 34, 43, 54mrcssd 16887 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
56 sslin 4161 . . . . . . . . 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})))))
5755, 56syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))))
581adantr 484 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐺dom DProd 𝑆)
594adantr 484 . . . . . . . . 9 ((𝜑𝑥𝐴) → dom 𝑆 = 𝐼)
606sselda 3915 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐼)
61 eqid 2798 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
6258, 59, 60, 61, 34dprddisj 19124 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)})
6357, 62sseqtrd 3955 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ {(0g𝐺)})
645ffvelrnda 6828 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
6560, 64syldan 594 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
6661subg0cl 18279 . . . . . . . . . 10 ((𝑆𝑥) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝑆𝑥))
6765, 66syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝑆𝑥))
6843, 54sstrd 3925 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺))
6934mrccl 16874 . . . . . . . . . . 11 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7033, 68, 69syl2anc 587 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7161subg0cl 18279 . . . . . . . . . 10 (((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))))
7270, 71syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))))
7367, 72elind 4121 . . . . . . . 8 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
7473snssd 4702 . . . . . . 7 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
7563, 74eqssd 3932 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)})
7628, 75eqtrd 2833 . . . . 5 ((𝜑𝑥𝐴) → (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)})
7725, 76jca 515 . . . 4 ((𝜑𝑥𝐴) → (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))
7877ralrimiva 3149 . . 3 (𝜑 → ∀𝑥𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))
791, 4dprddomcld 19116 . . . . 5 (𝜑𝐼 ∈ V)
8079, 6ssexd 5192 . . . 4 (𝜑𝐴 ∈ V)
817fdmd 6497 . . . 4 (𝜑 → dom (𝑆𝐴) = 𝐴)
8219, 61, 34dmdprd 19113 . . . 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)}))))
8380, 81, 82syl2anc 587 . . 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)}))))
843, 7, 78, 83mpbir3and 1339 . 2 (𝜑𝐺dom DProd (𝑆𝐴))
85 rnss 5773 . . . . . 6 ((𝑆𝐴) ⊆ 𝑆 → ran (𝑆𝐴) ⊆ ran 𝑆)
86 uniss 4808 . . . . . 6 (ran (𝑆𝐴) ⊆ ran 𝑆 ran (𝑆𝐴) ⊆ ran 𝑆)
8735, 85, 86mp2b 10 . . . . 5 ran (𝑆𝐴) ⊆ ran 𝑆
8887a1i 11 . . . 4 (𝜑 ran (𝑆𝐴) ⊆ ran 𝑆)
89 sspwuni 4985 . . . . 5 (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ran 𝑆 ⊆ (Base‘𝐺))
9050, 89sylib 221 . . . 4 (𝜑 ran 𝑆 ⊆ (Base‘𝐺))
9132, 34, 88, 90mrcssd 16887 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran (𝑆𝐴)) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
9234dprdspan 19142 . . . 4 (𝐺dom DProd (𝑆𝐴) → (𝐺 DProd (𝑆𝐴)) = ((mrCls‘(SubGrp‘𝐺))‘ ran (𝑆𝐴)))
9384, 92syl 17 . . 3 (𝜑 → (𝐺 DProd (𝑆𝐴)) = ((mrCls‘(SubGrp‘𝐺))‘ ran (𝑆𝐴)))
9434dprdspan 19142 . . . 4 (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
951, 94syl 17 . . 3 (𝜑 → (𝐺 DProd 𝑆) = ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
9691, 93, 953sstr4d 3962 . 2 (𝜑 → (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆))
9784, 96jca 515 1 (𝜑 → (𝐺dom DProd (𝑆𝐴) ∧ (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wne 2987  wral 3106  Vcvv 3441  cdif 3878  cin 3880  wss 3881  𝒫 cpw 4497  {csn 4525   cuni 4800   class class class wbr 5030  dom cdm 5519  ran crn 5520  cres 5521  cima 5522  wf 6320  cfv 6324  (class class class)co 7135  Basecbs 16475  0gc0g 16705  Moorecmre 16845  mrClscmrc 16846  ACScacs 16848  Grpcgrp 18095  SubGrpcsubg 18265  Cntzccntz 18437   DProd cdprd 19108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-tpos 7875  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-ghm 18348  df-gim 18391  df-cntz 18439  df-oppg 18466  df-cmn 18900  df-dprd 19110
This theorem is referenced by:  dprdf1  19148  dprdcntz2  19153  dprddisj2  19154  dprd2dlem1  19156  dprd2da  19157  dmdprdsplit  19162  dprdsplit  19163  dpjf  19172  dpjidcl  19173  dpjlid  19176  dpjghm  19178  ablfac1eulem  19187  ablfac1eu  19188
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