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Mirrors > Home > MPE Home > Th. List > dprdspan | Structured version Visualization version GIF version |
Description: The direct product is the span of the union of the factors. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
dprdspan.k | ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) |
Ref | Expression |
---|---|
dprdspan | ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = (𝐾‘∪ ran 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝐺dom DProd 𝑆 → 𝐺dom DProd 𝑆) | |
2 | eqidd 2822 | . . 3 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 = dom 𝑆) | |
3 | dprdgrp 19121 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
4 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 4 | subgacs 18307 | . . . . 5 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺))) |
6 | acsmre 16917 | . . . . 5 ⊢ ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺))) | |
7 | 3, 5, 6 | 3syl 18 | . . . 4 ⊢ (𝐺dom DProd 𝑆 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺))) |
8 | dprdf 19122 | . . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺)) | |
9 | 8 | ffnd 6509 | . . . . . . 7 ⊢ (𝐺dom DProd 𝑆 → 𝑆 Fn dom 𝑆) |
10 | fniunfv 7000 | . . . . . . 7 ⊢ (𝑆 Fn dom 𝑆 → ∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) = ∪ ran 𝑆) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝐺dom DProd 𝑆 → ∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) = ∪ ran 𝑆) |
12 | simpl 485 | . . . . . . . . 9 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → 𝐺dom DProd 𝑆) | |
13 | eqidd 2822 | . . . . . . . . 9 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → dom 𝑆 = dom 𝑆) | |
14 | simpr 487 | . . . . . . . . 9 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → 𝑘 ∈ dom 𝑆) | |
15 | 12, 13, 14 | dprdub 19141 | . . . . . . . 8 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → (𝑆‘𝑘) ⊆ (𝐺 DProd 𝑆)) |
16 | 15 | ralrimiva 3182 | . . . . . . 7 ⊢ (𝐺dom DProd 𝑆 → ∀𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ (𝐺 DProd 𝑆)) |
17 | iunss 4961 | . . . . . . 7 ⊢ (∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ (𝐺 DProd 𝑆) ↔ ∀𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ (𝐺 DProd 𝑆)) | |
18 | 16, 17 | sylibr 236 | . . . . . 6 ⊢ (𝐺dom DProd 𝑆 → ∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ (𝐺 DProd 𝑆)) |
19 | 11, 18 | eqsstrrd 4005 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → ∪ ran 𝑆 ⊆ (𝐺 DProd 𝑆)) |
20 | 4 | dprdssv 19132 | . . . . 5 ⊢ (𝐺 DProd 𝑆) ⊆ (Base‘𝐺) |
21 | 19, 20 | sstrdi 3978 | . . . 4 ⊢ (𝐺dom DProd 𝑆 → ∪ ran 𝑆 ⊆ (Base‘𝐺)) |
22 | dprdspan.k | . . . . 5 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) | |
23 | 22 | mrccl 16876 | . . . 4 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ (Base‘𝐺)) → (𝐾‘∪ ran 𝑆) ∈ (SubGrp‘𝐺)) |
24 | 7, 21, 23 | syl2anc 586 | . . 3 ⊢ (𝐺dom DProd 𝑆 → (𝐾‘∪ ran 𝑆) ∈ (SubGrp‘𝐺)) |
25 | eqimss 4022 | . . . . . . 7 ⊢ (∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) = ∪ ran 𝑆 → ∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ ∪ ran 𝑆) | |
26 | 11, 25 | syl 17 | . . . . . 6 ⊢ (𝐺dom DProd 𝑆 → ∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ ∪ ran 𝑆) |
27 | iunss 4961 | . . . . . 6 ⊢ (∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ ∪ ran 𝑆 ↔ ∀𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ ∪ ran 𝑆) | |
28 | 26, 27 | sylib 220 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → ∀𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ ∪ ran 𝑆) |
29 | 28 | r19.21bi 3208 | . . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → (𝑆‘𝑘) ⊆ ∪ ran 𝑆) |
30 | 7, 22, 21 | mrcssidd 16890 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → ∪ ran 𝑆 ⊆ (𝐾‘∪ ran 𝑆)) |
31 | 30 | adantr 483 | . . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → ∪ ran 𝑆 ⊆ (𝐾‘∪ ran 𝑆)) |
32 | 29, 31 | sstrd 3976 | . . 3 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → (𝑆‘𝑘) ⊆ (𝐾‘∪ ran 𝑆)) |
33 | 1, 2, 24, 32 | dprdlub 19142 | . 2 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ⊆ (𝐾‘∪ ran 𝑆)) |
34 | dprdsubg 19140 | . . 3 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺)) | |
35 | 22 | mrcsscl 16885 | . . 3 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ (𝐺 DProd 𝑆) ∧ (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺)) → (𝐾‘∪ ran 𝑆) ⊆ (𝐺 DProd 𝑆)) |
36 | 7, 19, 34, 35 | syl3anc 1367 | . 2 ⊢ (𝐺dom DProd 𝑆 → (𝐾‘∪ ran 𝑆) ⊆ (𝐺 DProd 𝑆)) |
37 | 33, 36 | eqssd 3983 | 1 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = (𝐾‘∪ ran 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 ∪ cuni 4831 ∪ ciun 4911 class class class wbr 5058 dom cdm 5549 ran crn 5550 Fn wfn 6344 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Moorecmre 16847 mrClscmrc 16848 ACScacs 16850 Grpcgrp 18097 SubGrpcsubg 18267 DProd cdprd 19109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-gim 18393 df-cntz 18441 df-oppg 18468 df-cmn 18902 df-dprd 19111 |
This theorem is referenced by: dprdres 19144 dprdf1o 19148 subgdprd 19151 dprdsn 19152 dprd2dlem1 19157 dprd2da 19158 dprd2db 19159 dmdprdsplit2lem 19161 |
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