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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 2823 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | eqid 2823 | . . . 4 ⊢ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} | |
5 | 3, 4 | dprdval 19127 | . . 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 586 | . 2 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} ↦ (𝐺 Σg 𝑓))) |
7 | eqid 2823 | . . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
8 | 1 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝐺dom DProd 𝑆) |
9 | dprdgrp 19129 | . . . . . 6 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
10 | grpmnd 18112 | . . . . . 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 19125 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
13 | 12 | adantr 483 | . . . . 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 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑇 ∈ (SubGrp‘𝐺)) |
16 | subgsubm 18303 | . . . . . 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 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → dom 𝑆 = 𝐼) |
19 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) | |
20 | eqid 2823 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
21 | 4, 8, 18, 19, 20 | dprdff 19136 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓:𝐼⟶(Base‘𝐺)) |
22 | 21 | ffnd 6517 | . . . . . 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 19137 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ (𝑆‘𝑘)) |
26 | 24, 25 | sseldd 3970 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ 𝑇) |
27 | 26 | ralrimiva 3184 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇) |
28 | ffnfv 6884 | . . . . . 6 ⊢ (𝑓:𝐼⟶𝑇 ↔ (𝑓 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑓‘𝑘) ∈ 𝑇)) | |
29 | 22, 27, 28 | sylanbrc 585 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓:𝐼⟶𝑇) |
30 | 4, 8, 18, 19, 7 | dprdfcntz 19139 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓)) |
31 | 4, 8, 18, 19 | dprdffsupp 19138 | . . . . 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 19040 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → (𝐺 Σg 𝑓) ∈ 𝑇) |
33 | 32 | fmpttd 6881 | . . 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 6523 | . 2 ⊢ (𝜑 → ran (𝑓 ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} ↦ (𝐺 Σg 𝑓)) ⊆ 𝑇) |
35 | 6, 34 | eqsstrd 4007 | 1 ⊢ (𝜑 → (𝐺 DProd 𝑆) ⊆ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 Vcvv 3496 ⊆ wss 3938 class class class wbr 5068 ↦ cmpt 5148 dom cdm 5557 ran crn 5558 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Xcixp 8463 finSupp cfsupp 8835 Basecbs 16485 0gc0g 16715 Σg cgsu 16716 Mndcmnd 17913 SubMndcsubmnd 17957 Grpcgrp 18105 SubGrpcsubg 18275 Cntzccntz 18447 DProd cdprd 19117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-0g 16717 df-gsum 16718 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-subg 18278 df-cntz 18449 df-dprd 19119 |
This theorem is referenced by: dprdspan 19151 dprdz 19154 dprdcntz2 19162 dprd2dlem1 19165 dprdsplit 19172 ablfac1eu 19197 |
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