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Mirrors > Home > MPE Home > Th. List > dprdssv | Structured version Visualization version GIF version |
Description: The internal direct product of a family of subgroups is a subset of the base. (Contributed by Mario Carneiro, 25-Apr-2016.) |
Ref | Expression |
---|---|
dprdssv.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
dprdssv | ⊢ (𝐺 DProd 𝑆) ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 ⊢ dom 𝑆 = dom 𝑆 | |
2 | eqid 2737 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | eqid 2737 | . . . . 5 ⊢ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} = {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)} | |
4 | 2, 3 | eldprd 19391 | . . . 4 ⊢ (dom 𝑆 = dom 𝑆 → (𝑥 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}𝑥 = (𝐺 Σg 𝑓)))) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ (𝑥 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}𝑥 = (𝐺 Σg 𝑓))) |
6 | dprdssv.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
7 | eqid 2737 | . . . . . . 7 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
8 | dprdgrp 19392 | . . . . . . . . 9 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
9 | 8 | grpmndd 18377 | . . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Mnd) |
10 | 9 | adantr 484 | . . . . . . 7 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝐺 ∈ Mnd) |
11 | reldmdprd 19384 | . . . . . . . . . 10 ⊢ Rel dom DProd | |
12 | 11 | brrelex2i 5606 | . . . . . . . . 9 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V) |
13 | 12 | dmexd 7683 | . . . . . . . 8 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V) |
14 | 13 | adantr 484 | . . . . . . 7 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → dom 𝑆 ∈ V) |
15 | simpl 486 | . . . . . . . 8 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝐺dom DProd 𝑆) | |
16 | eqidd 2738 | . . . . . . . 8 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → dom 𝑆 = dom 𝑆) | |
17 | simpr 488 | . . . . . . . 8 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) | |
18 | 3, 15, 16, 17, 6 | dprdff 19399 | . . . . . . 7 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓:dom 𝑆⟶𝐵) |
19 | 3, 15, 16, 17, 7 | dprdfcntz 19402 | . . . . . . 7 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓)) |
20 | 3, 15, 16, 17 | dprdffsupp 19401 | . . . . . . 7 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → 𝑓 finSupp (0g‘𝐺)) |
21 | 6, 2, 7, 10, 14, 18, 19, 20 | gsumzcl 19296 | . . . . . 6 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → (𝐺 Σg 𝑓) ∈ 𝐵) |
22 | eleq1 2825 | . . . . . 6 ⊢ (𝑥 = (𝐺 Σg 𝑓) → (𝑥 ∈ 𝐵 ↔ (𝐺 Σg 𝑓) ∈ 𝐵)) | |
23 | 21, 22 | syl5ibrcom 250 | . . . . 5 ⊢ ((𝐺dom DProd 𝑆 ∧ 𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}) → (𝑥 = (𝐺 Σg 𝑓) → 𝑥 ∈ 𝐵)) |
24 | 23 | rexlimdva 3203 | . . . 4 ⊢ (𝐺dom DProd 𝑆 → (∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}𝑥 = (𝐺 Σg 𝑓) → 𝑥 ∈ 𝐵)) |
25 | 24 | imp 410 | . . 3 ⊢ ((𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ {ℎ ∈ X𝑖 ∈ dom 𝑆(𝑆‘𝑖) ∣ ℎ finSupp (0g‘𝐺)}𝑥 = (𝐺 Σg 𝑓)) → 𝑥 ∈ 𝐵) |
26 | 5, 25 | sylbi 220 | . 2 ⊢ (𝑥 ∈ (𝐺 DProd 𝑆) → 𝑥 ∈ 𝐵) |
27 | 26 | ssriv 3905 | 1 ⊢ (𝐺 DProd 𝑆) ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3062 {crab 3065 Vcvv 3408 ⊆ wss 3866 class class class wbr 5053 dom cdm 5551 ‘cfv 6380 (class class class)co 7213 Xcixp 8578 finSupp cfsupp 8985 Basecbs 16760 0gc0g 16944 Σg cgsu 16945 Mndcmnd 18173 Cntzccntz 18709 DProd cdprd 19380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-fzo 13239 df-seq 13575 df-hash 13897 df-0g 16946 df-gsum 16947 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-subg 18540 df-cntz 18711 df-dprd 19382 |
This theorem is referenced by: dprdfsub 19408 dprdf11 19410 dprdsubg 19411 dprdspan 19414 dprdcntz2 19425 dprd2da 19429 dmdprdsplit2lem 19432 ablfac1c 19458 ablfac1eulem 19459 ablfac1eu 19460 |
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