![]() |
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 2736 | . 2 ⊢ (𝜑 → (Base‘𝑌) = (Base‘𝑌)) | |
2 | eqidd 2736 | . 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 18971 | . . . . 5 ⊢ (𝑎 ∈ Grp → 𝑎 ∈ Mnd) | |
8 | 7 | ssriv 3999 | . . . 4 ⊢ Grp ⊆ Mnd |
9 | fss 6753 | . . . 4 ⊢ ((𝑅:𝐼⟶Grp ∧ Grp ⊆ Mnd) → 𝑅:𝐼⟶Mnd) | |
10 | 6, 8, 9 | sylancl 586 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
11 | 3, 4, 5, 10 | prds0g 18797 | . 2 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
12 | 3, 4, 5, 10 | prdsmndd 18796 | . 2 ⊢ (𝜑 → 𝑌 ∈ Mnd) |
13 | eqid 2735 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
14 | eqid 2735 | . . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
15 | 5 | elexd 3502 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ V) |
16 | 15 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑆 ∈ V) |
17 | 4 | elexd 3502 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
18 | 17 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝐼 ∈ V) |
19 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp) |
20 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → 𝑎 ∈ (Base‘𝑌)) | |
21 | eqid 2735 | . . . 4 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅) | |
22 | eqid 2735 | . . . 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 19080 | . . 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 494 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘𝑌)) → (𝑏 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑏))‘(𝑎‘𝑏))) ∈ (Base‘𝑌)) |
25 | 23 | simprd 495 | . 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 18987 | 1 ⊢ (𝜑 → 𝑌 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ↦ cmpt 5231 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 Xscprds 17492 Mndcmnd 18760 Grpcgrp 18964 invgcminusg 18965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 |
This theorem is referenced by: prdsinvgd 19082 pwsgrp 19083 xpsgrp 19090 prdsabld 19895 prdsringd 20335 prdslmodd 20985 dsmmsubg 21781 prdstgpd 24149 |
Copyright terms: Public domain | W3C validator |