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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 2728 | . . 3 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 = dom 𝑆) | |
3 | dprdgrp 19967 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
4 | eqid 2727 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
5 | 4 | subgacs 19121 | . . . . 5 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺))) |
6 | acsmre 17637 | . . . . 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 19968 | . . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺)) | |
9 | 8 | ffnd 6726 | . . . . . . 7 ⊢ (𝐺dom DProd 𝑆 → 𝑆 Fn dom 𝑆) |
10 | fniunfv 7261 | . . . . . . 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 481 | . . . . . . . . 9 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → 𝐺dom DProd 𝑆) | |
13 | eqidd 2728 | . . . . . . . . 9 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → dom 𝑆 = dom 𝑆) | |
14 | simpr 483 | . . . . . . . . 9 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → 𝑘 ∈ dom 𝑆) | |
15 | 12, 13, 14 | dprdub 19987 | . . . . . . . 8 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → (𝑆‘𝑘) ⊆ (𝐺 DProd 𝑆)) |
16 | 15 | ralrimiva 3142 | . . . . . . 7 ⊢ (𝐺dom DProd 𝑆 → ∀𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ (𝐺 DProd 𝑆)) |
17 | iunss 5050 | . . . . . . 7 ⊢ (∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ (𝐺 DProd 𝑆) ↔ ∀𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ (𝐺 DProd 𝑆)) | |
18 | 16, 17 | sylibr 233 | . . . . . 6 ⊢ (𝐺dom DProd 𝑆 → ∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ (𝐺 DProd 𝑆)) |
19 | 11, 18 | eqsstrrd 4019 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → ∪ ran 𝑆 ⊆ (𝐺 DProd 𝑆)) |
20 | 4 | dprdssv 19978 | . . . . 5 ⊢ (𝐺 DProd 𝑆) ⊆ (Base‘𝐺) |
21 | 19, 20 | sstrdi 3992 | . . . 4 ⊢ (𝐺dom DProd 𝑆 → ∪ ran 𝑆 ⊆ (Base‘𝐺)) |
22 | dprdspan.k | . . . . 5 ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺)) | |
23 | 22 | mrccl 17596 | . . . 4 ⊢ (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ∪ ran 𝑆 ⊆ (Base‘𝐺)) → (𝐾‘∪ ran 𝑆) ∈ (SubGrp‘𝐺)) |
24 | 7, 21, 23 | syl2anc 582 | . . 3 ⊢ (𝐺dom DProd 𝑆 → (𝐾‘∪ ran 𝑆) ∈ (SubGrp‘𝐺)) |
25 | eqimss 4038 | . . . . . . 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 5050 | . . . . . 6 ⊢ (∪ 𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ ∪ ran 𝑆 ↔ ∀𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ ∪ ran 𝑆) | |
28 | 26, 27 | sylib 217 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → ∀𝑘 ∈ dom 𝑆(𝑆‘𝑘) ⊆ ∪ ran 𝑆) |
29 | 28 | r19.21bi 3244 | . . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → (𝑆‘𝑘) ⊆ ∪ ran 𝑆) |
30 | 7, 22, 21 | mrcssidd 17610 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → ∪ ran 𝑆 ⊆ (𝐾‘∪ ran 𝑆)) |
31 | 30 | adantr 479 | . . . 4 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → ∪ ran 𝑆 ⊆ (𝐾‘∪ ran 𝑆)) |
32 | 29, 31 | sstrd 3990 | . . 3 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑘 ∈ dom 𝑆) → (𝑆‘𝑘) ⊆ (𝐾‘∪ ran 𝑆)) |
33 | 1, 2, 24, 32 | dprdlub 19988 | . 2 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ⊆ (𝐾‘∪ ran 𝑆)) |
34 | dprdsubg 19986 | . . 3 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) ∈ (SubGrp‘𝐺)) | |
35 | 22 | mrcsscl 17605 | . . 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 1368 | . 2 ⊢ (𝐺dom DProd 𝑆 → (𝐾‘∪ ran 𝑆) ⊆ (𝐺 DProd 𝑆)) |
37 | 33, 36 | eqssd 3997 | 1 ⊢ (𝐺dom DProd 𝑆 → (𝐺 DProd 𝑆) = (𝐾‘∪ ran 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3057 ⊆ wss 3947 ∪ cuni 4910 ∪ ciun 4998 class class class wbr 5150 dom cdm 5680 ran crn 5681 Fn wfn 6546 ‘cfv 6551 (class class class)co 7424 Basecbs 17185 Moorecmre 17567 mrClscmrc 17568 ACScacs 17570 Grpcgrp 18895 SubGrpcsubg 19080 DProd cdprd 19955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-tpos 8236 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-map 8851 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-n0 12509 df-z 12595 df-uz 12859 df-fz 13523 df-fzo 13666 df-seq 14005 df-hash 14328 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-0g 17428 df-gsum 17429 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18745 df-submnd 18746 df-grp 18898 df-minusg 18899 df-sbg 18900 df-mulg 19029 df-subg 19083 df-ghm 19173 df-gim 19218 df-cntz 19273 df-oppg 19302 df-cmn 19742 df-dprd 19957 |
This theorem is referenced by: dprdres 19990 dprdf1o 19994 subgdprd 19997 dprdsn 19998 dprd2dlem1 20003 dprd2da 20004 dprd2db 20005 dmdprdsplit2lem 20007 |
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