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Mirrors > Home > MPE Home > Th. List > prdsgrpd | Structured version Visualization version GIF version |
Description: The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
prdsgrpd.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsgrpd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsgrpd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsgrpd.r | ⊢ (𝜑 → 𝑅:𝐼⟶Grp) |
Ref | Expression |
---|---|
prdsgrpd | ⊢ (𝜑 → 𝑌 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2826 | . 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌)) | |
2 | eqidd 2826 | . 2 ⊢ (𝜑 → (+g‘𝑌) = (+g‘𝑌)) | |
3 | prdsgrpd.y | . . 3 ⊢ 𝑌 = (𝑆Xs𝑅) | |
4 | prdsgrpd.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | prdsgrpd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
6 | prdsgrpd.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Grp) | |
7 | grpmnd 17790 | . . . . 5 ⊢ (𝑎 ∈ Grp → 𝑎 ∈ Mnd) | |
8 | 7 | ssriv 3831 | . . . 4 ⊢ Grp ⊆ Mnd |
9 | fss 6295 | . . . 4 ⊢ ((𝑅:𝐼⟶Grp ∧ Grp ⊆ Mnd) → 𝑅:𝐼⟶Mnd) | |
10 | 6, 8, 9 | sylancl 580 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
11 | 3, 4, 5, 10 | prds0g 17684 | . 2 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
12 | 3, 4, 5, 10 | prdsmndd 17683 | . 2 ⊢ (𝜑 → 𝑌 ∈ Mnd) |
13 | eqid 2825 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
14 | eqid 2825 | . . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
15 | elex 3429 | . . . . . 6 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
16 | 5, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ V) |
17 | 16 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ V) |
18 | elex 3429 | . . . . . 6 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
19 | 4, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
20 | 19 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V) |
21 | 6 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp) |
22 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌)) | |
23 | eqid 2825 | . . . 4 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅) | |
24 | eqid 2825 | . . . 4 ⊢ (𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏))) = (𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏))) | |
25 | 3, 13, 14, 17, 20, 21, 22, 23, 24 | prdsinvlem 17885 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏))) ∈ (Base‘𝑌) ∧ ((𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏)))(+g‘𝑌)𝑎) = (0g ∘ 𝑅))) |
26 | 25 | simpld 490 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏))) ∈ (Base‘𝑌)) |
27 | 25 | simprd 491 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏)))(+g‘𝑌)𝑎) = (0g ∘ 𝑅)) |
28 | 1, 2, 11, 12, 26, 27 | isgrpd2 17803 | 1 ⊢ (𝜑 → 𝑌 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ⊆ wss 3798 ↦ cmpt 4954 ∘ ccom 5350 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 0gc0g 16460 Xscprds 16466 Mndcmnd 17654 Grpcgrp 17783 invgcminusg 17784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-hom 16336 df-cco 16337 df-0g 16462 df-prds 16468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 |
This theorem is referenced by: prdsinvgd 17887 pwsgrp 17888 xpsgrp 17895 prdsabld 18625 prdsringd 18973 prdslmodd 19335 dsmmsubg 20457 prdstgpd 22305 |
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