MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprdres Structured version   Visualization version   GIF version

Theorem dprdres 19881
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 19858 . . . 4 (𝐺dom DProd 𝑆𝐺 ∈ Grp)
31, 2syl 17 . . 3 (𝜑𝐺 ∈ Grp)
4 dprdres.2 . . . . 5 (𝜑 → dom 𝑆 = 𝐼)
51, 4dprdf2 19860 . . . 4 (𝜑𝑆:𝐼⟶(SubGrp‘𝐺))
6 dprdres.3 . . . 4 (𝜑𝐴𝐼)
75, 6fssresd 6748 . . 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 3981 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝐼)
13 eldifi 4124 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦𝐴)
1413adantl 483 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝐴)
1510, 14sseldd 3981 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝐼)
16 eldifsni 4789 . . . . . . . . . 10 (𝑦 ∈ (𝐴 ∖ {𝑥}) → 𝑦𝑥)
1716adantl 483 . . . . . . . . 9 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑦𝑥)
1817necomd 2997 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → 𝑥𝑦)
19 eqid 2733 . . . . . . . 8 (Cntz‘𝐺) = (Cntz‘𝐺)
208, 9, 12, 15, 18, 19dprdcntz 19861 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
2111fvresd 6901 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2214fvresd 6901 . . . . . . . 8 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑦) = (𝑆𝑦))
2322fveq2d 6885 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) = ((Cntz‘𝐺)‘(𝑆𝑦)))
2420, 21, 233sstr4d 4027 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝑦 ∈ (𝐴 ∖ {𝑥})) → ((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)))
2524ralrimiva 3147 . . . . 5 ((𝜑𝑥𝐴) → ∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)))
26 fvres 6900 . . . . . . . 8 (𝑥𝐴 → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2726adantl 483 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝐴)‘𝑥) = (𝑆𝑥))
2827ineq1d 4209 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
29 eqid 2733 . . . . . . . . . . . . 13 (Base‘𝐺) = (Base‘𝐺)
3029subgacs 19026 . . . . . . . . . . . 12 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
31 acsmre 17583 . . . . . . . . . . . 12 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
323, 30, 313syl 18 . . . . . . . . . . 11 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
3332adantr 482 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
34 eqid 2733 . . . . . . . . . 10 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
35 resss 6001 . . . . . . . . . . . . 13 (𝑆𝐴) ⊆ 𝑆
36 imass1 6092 . . . . . . . . . . . . 13 ((𝑆𝐴) ⊆ 𝑆 → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥})))
3735, 36ax-mp 5 . . . . . . . . . . . 12 ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐴 ∖ {𝑥}))
386adantr 482 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → 𝐴𝐼)
39 ssdif 4137 . . . . . . . . . . . . 13 (𝐴𝐼 → (𝐴 ∖ {𝑥}) ⊆ (𝐼 ∖ {𝑥}))
40 imass2 6093 . . . . . . . . . . . . 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 3991 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
4342unissd 4914 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (𝑆 “ (𝐼 ∖ {𝑥})))
44 imassrn 6063 . . . . . . . . . . . 12 (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝑆
455frnd 6715 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
4629subgss 18992 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ (Base‘𝐺))
47 velpw 4603 . . . . . . . . . . . . . . . 16 (𝑥 ∈ 𝒫 (Base‘𝐺) ↔ 𝑥 ⊆ (Base‘𝐺))
4846, 47sylibr 233 . . . . . . . . . . . . . . 15 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ∈ 𝒫 (Base‘𝐺))
4948ssriv 3984 . . . . . . . . . . . . . 14 (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺)
5045, 49sstrdi 3992 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5150adantr 482 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5244, 51sstrid 3991 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
53 sspwuni 5099 . . . . . . . . . . 11 ((𝑆 “ (𝐼 ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
5452, 53sylib 217 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ (𝐼 ∖ {𝑥})) ⊆ (Base‘𝐺))
5533, 34, 43, 54mrcssd 17555 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥}))))
56 sslin 4232 . . . . . . . . 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 482 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝐺dom DProd 𝑆)
594adantr 482 . . . . . . . . 9 ((𝜑𝑥𝐴) → dom 𝑆 = 𝐼)
606sselda 3980 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐼)
61 eqid 2733 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
6258, 59, 60, 61, 34dprddisj 19862 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (𝐼 ∖ {𝑥})))) = {(0g𝐺)})
6357, 62sseqtrd 4020 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) ⊆ {(0g𝐺)})
645ffvelcdmda 7074 . . . . . . . . . . 11 ((𝜑𝑥𝐼) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
6560, 64syldan 592 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
6661subg0cl 18999 . . . . . . . . . 10 ((𝑆𝑥) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝑆𝑥))
6765, 66syl 17 . . . . . . . . 9 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝑆𝑥))
6843, 54sstrd 3990 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺))
6934mrccl 17542 . . . . . . . . . . 11 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})) ⊆ (Base‘𝐺)) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7033, 68, 69syl2anc 585 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
7161subg0cl 18999 . . . . . . . . . 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 4192 . . . . . . . 8 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
7473snssd 4808 . . . . . . 7 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))))
7563, 74eqssd 3997 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)})
7628, 75eqtrd 2773 . . . . 5 ((𝜑𝑥𝐴) → (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)})
7725, 76jca 513 . . . 4 ((𝜑𝑥𝐴) → (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))
7877ralrimiva 3147 . . 3 (𝜑 → ∀𝑥𝐴 (∀𝑦 ∈ (𝐴 ∖ {𝑥})((𝑆𝐴)‘𝑥) ⊆ ((Cntz‘𝐺)‘((𝑆𝐴)‘𝑦)) ∧ (((𝑆𝐴)‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ ((𝑆𝐴) “ (𝐴 ∖ {𝑥})))) = {(0g𝐺)}))
791, 4dprddomcld 19854 . . . . 5 (𝜑𝐼 ∈ V)
8079, 6ssexd 5320 . . . 4 (𝜑𝐴 ∈ V)
817fdmd 6718 . . . 4 (𝜑 → dom (𝑆𝐴) = 𝐴)
8219, 61, 34dmdprd 19851 . . . 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 585 . . 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 1343 . 2 (𝜑𝐺dom DProd (𝑆𝐴))
85 rnss 5933 . . . . . 6 ((𝑆𝐴) ⊆ 𝑆 → ran (𝑆𝐴) ⊆ ran 𝑆)
86 uniss 4912 . . . . . 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 5099 . . . . 5 (ran 𝑆 ⊆ 𝒫 (Base‘𝐺) ↔ ran 𝑆 ⊆ (Base‘𝐺))
9050, 89sylib 217 . . . 4 (𝜑 ran 𝑆 ⊆ (Base‘𝐺))
9132, 34, 88, 90mrcssd 17555 . . 3 (𝜑 → ((mrCls‘(SubGrp‘𝐺))‘ ran (𝑆𝐴)) ⊆ ((mrCls‘(SubGrp‘𝐺))‘ ran 𝑆))
9234dprdspan 19880 . . . 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 19880 . . . 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 4027 . 2 (𝜑 → (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆))
9784, 96jca 513 1 (𝜑 → (𝐺dom DProd (𝑆𝐴) ∧ (𝐺 DProd (𝑆𝐴)) ⊆ (𝐺 DProd 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  Vcvv 3475  cdif 3943  cin 3945  wss 3946  𝒫 cpw 4598  {csn 4624   cuni 4904   class class class wbr 5144  dom cdm 5672  ran crn 5673  cres 5674  cima 5675  wf 6531  cfv 6535  (class class class)co 7396  Basecbs 17131  0gc0g 17372  Moorecmre 17513  mrClscmrc 17514  ACScacs 17516  Grpcgrp 18806  SubGrpcsubg 18985  Cntzccntz 19164   DProd cdprd 19846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-iin 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7657  df-om 7843  df-1st 7962  df-2nd 7963  df-supp 8134  df-tpos 8198  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8691  df-map 8810  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9350  df-oi 9492  df-card 9921  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-nn 12200  df-2 12262  df-n0 12460  df-z 12546  df-uz 12810  df-fz 13472  df-fzo 13615  df-seq 13954  df-hash 14278  df-sets 17084  df-slot 17102  df-ndx 17114  df-base 17132  df-ress 17161  df-plusg 17197  df-0g 17374  df-gsum 17375  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18548  df-sgrp 18597  df-mnd 18613  df-mhm 18658  df-submnd 18659  df-grp 18809  df-minusg 18810  df-sbg 18811  df-mulg 18936  df-subg 18988  df-ghm 19075  df-gim 19118  df-cntz 19166  df-oppg 19194  df-cmn 19634  df-dprd 19848
This theorem is referenced by:  dprdf1  19886  dprdcntz2  19891  dprddisj2  19892  dprd2dlem1  19894  dprd2da  19895  dmdprdsplit  19900  dprdsplit  19901  dpjf  19910  dpjidcl  19911  dpjlid  19914  dpjghm  19916  ablfac1eulem  19925  ablfac1eu  19926
  Copyright terms: Public domain W3C validator