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Mirrors > Home > MPE Home > Th. List > dprdlub | Structured version Visualization version GIF version |
Description: The direct product is smaller than any subgroup which contains the factors. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
dprdlub.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dprdlub.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dprdlub.3 | ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) |
dprdlub.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇) |
Ref | Expression |
---|---|
dprdlub | ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dprdlub.1 | . . 3 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | dprdlub.2 | . . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
3 | eqid 2799 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | eqid 2799 | . . . 4 ⊢ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} | |
5 | 3, 4 | dprdval 18718 | . . 3 ⊢ ((𝐺dom DProd 𝑆 ∧ dom 𝑆 = 𝐼) → (𝐺 DProd 𝑆) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} ↦ (𝐺 Σg 𝑓))) |
6 | 1, 2, 5 | syl2anc 580 | . 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} ↦ (𝐺 Σg 𝑓))) |
7 | eqid 2799 | . . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
8 | 1 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝐺dom DProd 𝑆) |
9 | dprdgrp 18720 | . . . . . 6 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
10 | grpmnd 17745 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
11 | 8, 9, 10 | 3syl 18 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝐺 ∈ Mnd) |
12 | 1, 2 | dprddomcld 18716 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
13 | 12 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝐼 ∈ V) |
14 | dprdlub.3 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) | |
15 | 14 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑇 ∈ (SubGrp‘𝐺)) |
16 | subgsubm 17929 | . . . . . 6 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ∈ (SubMnd‘𝐺)) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑇 ∈ (SubMnd‘𝐺)) |
18 | 2 | adantr 473 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → dom 𝑆 = 𝐼) |
19 | simpr 478 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) | |
20 | eqid 2799 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
21 | 4, 8, 18, 19, 20 | dprdff 18727 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓:𝐼⟶(Base‘𝐺)) |
22 | 21 | ffnd 6257 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓 Fn 𝐼) |
23 | dprdlub.4 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇) | |
24 | 23 | adantlr 707 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇) |
25 | 4, 8, 18, 19 | dprdfcl 18728 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ (𝑆‘𝑘)) |
26 | 24, 25 | sseldd 3799 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ 𝑇) |
27 | 26 | ralrimiva 3147 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇) |
28 | ffnfv 6614 | . . . . . 6 ⊢ (𝑓:𝐼⟶𝑇 ↔ (𝑓 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇)) | |
29 | 22, 27, 28 | sylanbrc 579 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓:𝐼⟶𝑇) |
30 | 4, 8, 18, 19, 7 | dprdfcntz 18730 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓)) |
31 | 4, 8, 18, 19 | dprdffsupp 18729 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓 finSupp (0g‘𝐺)) |
32 | 3, 7, 11, 13, 17, 29, 30, 31 | gsumzsubmcl 18633 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → (𝐺 Σg 𝑓) ∈ 𝑇) |
33 | 32 | fmpttd 6611 | . . 3 ⊢ (𝜑 → (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} ↦ (𝐺 Σg 𝑓)):{ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}⟶𝑇) |
34 | 33 | frnd 6263 | . 2 ⊢ (𝜑 → ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} ↦ (𝐺 Σg 𝑓)) ⊆ 𝑇) |
35 | 6, 34 | eqsstrd 3835 | 1 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 {crab 3093 Vcvv 3385 ⊆ wss 3769 class class class wbr 4843 ↦ cmpt 4922 dom cdm 5312 ran crn 5313 Fn wfn 6096 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 Xcixp 8148 finSupp cfsupp 8517 Basecbs 16184 0gc0g 16415 Σg cgsu 16416 Mndcmnd 17609 SubMndcsubmnd 17649 Grpcgrp 17738 SubGrpcsubg 17901 Cntzccntz 18060 DProd cdprd 18708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-oi 8657 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-seq 13056 df-hash 13371 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-0g 16417 df-gsum 16418 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-grp 17741 df-minusg 17742 df-subg 17904 df-cntz 18062 df-dprd 18710 |
This theorem is referenced by: dprdspan 18742 dprdz 18745 dprdcntz2 18753 dprd2dlem1 18756 dprdsplit 18763 ablfac1eu 18788 |
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