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Mirrors > Home > MPE Home > Th. List > prdsabld | Structured version Visualization version GIF version |
Description: The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
prdscmnd.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdscmnd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdscmnd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsgabld.r | ⊢ (𝜑 → 𝑅:𝐼⟶Abel) |
Ref | Expression |
---|---|
prdsabld | ⊢ (𝜑 → 𝑌 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdscmnd.y | . . 3 ⊢ 𝑌 = (𝑆Xs𝑅) | |
2 | prdscmnd.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
3 | prdscmnd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | prdsgabld.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Abel) | |
5 | ablgrp 19783 | . . . . 5 ⊢ (𝑎 ∈ Abel → 𝑎 ∈ Grp) | |
6 | 5 | ssriv 3983 | . . . 4 ⊢ Abel ⊆ Grp |
7 | fss 6744 | . . . 4 ⊢ ((𝑅:𝐼⟶Abel ∧ Abel ⊆ Grp) → 𝑅:𝐼⟶Grp) | |
8 | 4, 6, 7 | sylancl 584 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp) |
9 | 1, 2, 3, 8 | prdsgrpd 19044 | . 2 ⊢ (𝜑 → 𝑌 ∈ Grp) |
10 | ablcmn 19785 | . . . . 5 ⊢ (𝑎 ∈ Abel → 𝑎 ∈ CMnd) | |
11 | 10 | ssriv 3983 | . . . 4 ⊢ Abel ⊆ CMnd |
12 | fss 6744 | . . . 4 ⊢ ((𝑅:𝐼⟶Abel ∧ Abel ⊆ CMnd) → 𝑅:𝐼⟶CMnd) | |
13 | 4, 11, 12 | sylancl 584 | . . 3 ⊢ (𝜑 → 𝑅:𝐼⟶CMnd) |
14 | 1, 2, 3, 13 | prdscmnd 19859 | . 2 ⊢ (𝜑 → 𝑌 ∈ CMnd) |
15 | isabl 19782 | . 2 ⊢ (𝑌 ∈ Abel ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ CMnd)) | |
16 | 9, 14, 15 | sylanbrc 581 | 1 ⊢ (𝜑 → 𝑌 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ⟶wf 6550 (class class class)co 7424 Xscprds 17460 Grpcgrp 18928 CMndccmn 19778 Abelcabl 19779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 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 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 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 12611 df-dec 12730 df-uz 12875 df-fz 13539 df-struct 17149 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-hom 17290 df-cco 17291 df-0g 17456 df-prds 17462 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-minusg 18932 df-cmn 19780 df-abl 19781 |
This theorem is referenced by: pwsabl 19862 prdsrngd 20159 |
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