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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 2799 | . 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌)) | |
2 | eqidd 2799 | . 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 18102 | . . . . 5 ⊢ (𝑎 ∈ Grp → 𝑎 ∈ Mnd) | |
8 | 7 | ssriv 3919 | . . . 4 ⊢ Grp ⊆ Mnd |
9 | fss 6501 | . . . 4 ⊢ ((𝑅:𝐼⟶Grp ∧ Grp ⊆ Mnd) → 𝑅:𝐼⟶Mnd) | |
10 | 6, 8, 9 | sylancl 589 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
11 | 3, 4, 5, 10 | prds0g 17937 | . 2 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
12 | 3, 4, 5, 10 | prdsmndd 17936 | . 2 ⊢ (𝜑 → 𝑌 ∈ Mnd) |
13 | eqid 2798 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
14 | eqid 2798 | . . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
15 | 5 | elexd 3461 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ V) |
16 | 15 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ V) |
17 | 4 | elexd 3461 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
18 | 17 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V) |
19 | 6 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp) |
20 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌)) | |
21 | eqid 2798 | . . . 4 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅) | |
22 | eqid 2798 | . . . 4 ⊢ (𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏))) = (𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏))) | |
23 | 3, 13, 14, 16, 18, 19, 20, 21, 22 | prdsinvlem 18200 | . . 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))) |
24 | 23 | simpld 498 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏))) ∈ (Base‘𝑌)) |
25 | 23 | simprd 499 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → ((𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏)))(+g‘𝑌)𝑎) = (0g ∘ 𝑅)) |
26 | 1, 2, 11, 12, 24, 25 | isgrpd2 18115 | 1 ⊢ (𝜑 → 𝑌 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ↦ cmpt 5110 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Xscprds 16711 Mndcmnd 17903 Grpcgrp 18095 invgcminusg 18096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-0g 16707 df-prds 16713 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 |
This theorem is referenced by: prdsinvgd 18202 pwsgrp 18203 xpsgrp 18210 prdsabld 18975 prdsringd 19358 prdslmodd 19734 dsmmsubg 20432 prdstgpd 22730 |
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