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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 2729 | . 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌)) | |
2 | eqidd 2729 | . 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 18904 | . . . . 5 ⊢ (𝑎 ∈ Grp → 𝑎 ∈ Mnd) | |
8 | 7 | ssriv 3986 | . . . 4 ⊢ Grp ⊆ Mnd |
9 | fss 6744 | . . . 4 ⊢ ((𝑅:𝐼⟶Grp ∧ Grp ⊆ Mnd) → 𝑅:𝐼⟶Mnd) | |
10 | 6, 8, 9 | sylancl 584 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
11 | 3, 4, 5, 10 | prds0g 18735 | . 2 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
12 | 3, 4, 5, 10 | prdsmndd 18734 | . 2 ⊢ (𝜑 → 𝑌 ∈ Mnd) |
13 | eqid 2728 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
14 | eqid 2728 | . . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
15 | 5 | elexd 3494 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ V) |
16 | 15 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ V) |
17 | 4 | elexd 3494 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
18 | 17 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V) |
19 | 6 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp) |
20 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌)) | |
21 | eqid 2728 | . . . 4 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅) | |
22 | eqid 2728 | . . . 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 19012 | . . 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 493 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏))) ∈ (Base‘𝑌)) |
25 | 23 | simprd 494 | . 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 18920 | 1 ⊢ (𝜑 → 𝑌 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⊆ wss 3949 ↦ cmpt 5235 ∘ ccom 5686 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 0gc0g 17428 Xscprds 17434 Mndcmnd 18701 Grpcgrp 18897 invgcminusg 18898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-prds 17436 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 |
This theorem is referenced by: prdsinvgd 19014 pwsgrp 19015 xpsgrp 19022 prdsabld 19824 prdsringd 20264 prdslmodd 20860 dsmmsubg 21684 prdstgpd 24049 |
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