Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2623 | . 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌)) | |
2 | eqidd 2623 | . 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 17429 | . . . . 5 ⊢ (𝑎 ∈ Grp → 𝑎 ∈ Mnd) | |
8 | 7 | ssriv 3607 | . . . 4 ⊢ Grp ⊆ Mnd |
9 | fss 6056 | . . . 4 ⊢ ((𝑅:𝐼⟶Grp ∧ Grp ⊆ Mnd) → 𝑅:𝐼⟶Mnd) | |
10 | 6, 8, 9 | sylancl 694 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
11 | 3, 4, 5, 10 | prds0g 17324 | . 2 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
12 | 3, 4, 5, 10 | prdsmndd 17323 | . 2 ⊢ (𝜑 → 𝑌 ∈ Mnd) |
13 | eqid 2622 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
14 | eqid 2622 | . . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
15 | elex 3212 | . . . . . 6 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
16 | 5, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ V) |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ V) |
18 | elex 3212 | . . . . . 6 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
19 | 4, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
20 | 19 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V) |
21 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp) |
22 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌)) | |
23 | eqid 2622 | . . . 4 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅) | |
24 | eqid 2622 | . . . 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 17524 | . . 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 475 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏))) ∈ (Base‘𝑌)) |
27 | 25 | simprd 479 | . 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 17442 | 1 ⊢ (𝜑 → 𝑌 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ↦ cmpt 4729 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 0gc0g 16100 Xscprds 16106 Mndcmnd 17294 Grpcgrp 17422 invgcminusg 17423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-prds 16108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 |
This theorem is referenced by: prdsinvgd 17526 pwsgrp 17527 xpsgrp 17534 prdsabld 18265 prdsringd 18612 prdslmodd 18969 dsmmsubg 20087 prdstgpd 21928 |
Copyright terms: Public domain | W3C validator |