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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 2725 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | eqid 2725 | . . . 4 ⊢ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} | |
5 | 3, 4 | dprdval 19972 | . . 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 582 | . 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} ↦ (𝐺 Σg 𝑓))) |
7 | eqid 2725 | . . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
8 | 1 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝐺dom DProd 𝑆) |
9 | dprdgrp 19974 | . . . . . 6 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
10 | grpmnd 18905 | . . . . . 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 19970 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
13 | 12 | adantr 479 | . . . . 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 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑇 ∈ (SubGrp‘𝐺)) |
16 | subgsubm 19111 | . . . . . 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 479 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → dom 𝑆 = 𝐼) |
19 | simpr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) | |
20 | eqid 2725 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
21 | 4, 8, 18, 19, 20 | dprdff 19981 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓:𝐼⟶(Base‘𝐺)) |
22 | 21 | ffnd 6724 | . . . . . 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 713 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑆‘𝑘) ⊆ 𝑇) |
25 | 4, 8, 18, 19 | dprdfcl 19982 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ (𝑆‘𝑘)) |
26 | 24, 25 | sseldd 3977 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ 𝑇) |
27 | 26 | ralrimiva 3135 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇) |
28 | ffnfv 7128 | . . . . . 6 ⊢ (𝑓:𝐼⟶𝑇 ↔ (𝑓 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇)) | |
29 | 22, 27, 28 | sylanbrc 581 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓:𝐼⟶𝑇) |
30 | 4, 8, 18, 19, 7 | dprdfcntz 19984 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓)) |
31 | 4, 8, 18, 19 | dprdffsupp 19983 | . . . . 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 19885 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → (𝐺 Σg 𝑓) ∈ 𝑇) |
33 | 32 | fmpttd 7124 | . . 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 6731 | . 2 ⊢ (𝜑 → ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} ↦ (𝐺 Σg 𝑓)) ⊆ 𝑇) |
35 | 6, 34 | eqsstrd 4015 | 1 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 {crab 3418 Vcvv 3461 ⊆ wss 3944 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5678 ran crn 5679 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 Xcixp 8916 finSupp cfsupp 9387 Basecbs 17183 0gc0g 17424 Σg cgsu 17425 Mndcmnd 18697 SubMndcsubmnd 18742 Grpcgrp 18898 SubGrpcsubg 19083 Cntzccntz 19278 DProd cdprd 19962 |
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 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-0g 17426 df-gsum 17427 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-subg 19086 df-cntz 19280 df-dprd 19964 |
This theorem is referenced by: dprdspan 19996 dprdz 19999 dprdcntz2 20007 dprd2dlem1 20010 dprdsplit 20017 ablfac1eu 20042 |
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