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Mirrors > Home > MPE Home > Th. List > dprdfid | Structured version Visualization version GIF version |
Description: A function mapping all but one arguments to zero sums to the value of this argument in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
Ref | Expression |
---|---|
eldprdi.0 | ⊢ 0 = (0g‘𝐺) |
eldprdi.w | ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
eldprdi.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
eldprdi.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dprdfid.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
dprdfid.4 | ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋)) |
dprdfid.f | ⊢ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) |
Ref | Expression |
---|---|
dprdfid | ⊢ (𝜑 → (𝐹 ∈ 𝑊 ∧ (𝐺 Σg 𝐹) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dprdfid.f | . . 3 ⊢ 𝐹 = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) | |
2 | eldprdi.w | . . . 4 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
3 | eldprdi.1 | . . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
4 | eldprdi.2 | . . . 4 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
5 | dprdfid.4 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋)) | |
6 | 5 | ad2antrr 718 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → 𝐴 ∈ (𝑆‘𝑋)) |
7 | simpr 478 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → 𝑛 = 𝑋) | |
8 | 7 | fveq2d 6416 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → (𝑆‘𝑛) = (𝑆‘𝑋)) |
9 | 6, 8 | eleqtrrd 2882 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → 𝐴 ∈ (𝑆‘𝑛)) |
10 | 3, 4 | dprdf2 18721 | . . . . . . . 8 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
11 | 10 | ffvelrnda 6586 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → (𝑆‘𝑛) ∈ (SubGrp‘𝐺)) |
12 | eldprdi.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝐺) | |
13 | 12 | subg0cl 17914 | . . . . . . 7 ⊢ ((𝑆‘𝑛) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆‘𝑛)) |
14 | 11, 13 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 0 ∈ (𝑆‘𝑛)) |
15 | 14 | adantr 473 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ ¬ 𝑛 = 𝑋) → 0 ∈ (𝑆‘𝑛)) |
16 | 9, 15 | ifclda 4312 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → if(𝑛 = 𝑋, 𝐴, 0 ) ∈ (𝑆‘𝑛)) |
17 | 3, 4 | dprddomcld 18715 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
18 | 12 | fvexi 6426 | . . . . . 6 ⊢ 0 ∈ V |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ V) |
20 | eqid 2800 | . . . . 5 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) | |
21 | 17, 19, 20 | sniffsupp 8558 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) finSupp 0 ) |
22 | 2, 3, 4, 16, 21 | dprdwd 18725 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) ∈ 𝑊) |
23 | 1, 22 | syl5eqel 2883 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
24 | eqid 2800 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
25 | dprdgrp 18719 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
26 | grpmnd 17744 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
27 | 3, 25, 26 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
28 | dprdfid.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
29 | 2, 3, 4, 23, 24 | dprdff 18726 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
30 | 1 | oveq1i 6889 | . . . . 5 ⊢ (𝐹 supp 0 ) = ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) |
31 | eldifsni 4511 | . . . . . . . 8 ⊢ (𝑛 ∈ (𝐼 ∖ {𝑋}) → 𝑛 ≠ 𝑋) | |
32 | 31 | adantl 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐼 ∖ {𝑋})) → 𝑛 ≠ 𝑋) |
33 | ifnefalse 4290 | . . . . . . 7 ⊢ (𝑛 ≠ 𝑋 → if(𝑛 = 𝑋, 𝐴, 0 ) = 0 ) | |
34 | 32, 33 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐼 ∖ {𝑋})) → if(𝑛 = 𝑋, 𝐴, 0 ) = 0 ) |
35 | 34, 17 | suppss2 7568 | . . . . 5 ⊢ (𝜑 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) ⊆ {𝑋}) |
36 | 30, 35 | syl5eqss 3846 | . . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋}) |
37 | 24, 12, 27, 17, 28, 29, 36 | gsumpt 18675 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐹‘𝑋)) |
38 | iftrue 4284 | . . . . 5 ⊢ (𝑛 = 𝑋 → if(𝑛 = 𝑋, 𝐴, 0 ) = 𝐴) | |
39 | 38, 1 | fvmptg 6506 | . . . 4 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝐴 ∈ (𝑆‘𝑋)) → (𝐹‘𝑋) = 𝐴) |
40 | 28, 5, 39 | syl2anc 580 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) = 𝐴) |
41 | 37, 40 | eqtrd 2834 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 𝐴) |
42 | 23, 41 | jca 508 | 1 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ∧ (𝐺 Σg 𝐹) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2972 {crab 3094 Vcvv 3386 ∖ cdif 3767 ifcif 4278 {csn 4369 class class class wbr 4844 ↦ cmpt 4923 dom cdm 5313 ‘cfv 6102 (class class class)co 6879 supp csupp 7533 Xcixp 8149 finSupp cfsupp 8518 Basecbs 16183 0gc0g 16414 Σg cgsu 16415 Mndcmnd 17608 Grpcgrp 17737 SubGrpcsubg 17900 DProd cdprd 18707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-inf2 8789 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-iin 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-se 5273 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-isom 6111 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-supp 7534 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-oadd 7804 df-er 7983 df-ixp 8150 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-fsupp 8519 df-oi 8658 df-card 9052 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-2 11375 df-n0 11580 df-z 11666 df-uz 11930 df-fz 12580 df-fzo 12720 df-seq 13055 df-hash 13370 df-ndx 16186 df-slot 16187 df-base 16189 df-sets 16190 df-ress 16191 df-plusg 16279 df-0g 16416 df-gsum 16417 df-mre 16560 df-mrc 16561 df-acs 16563 df-mgm 17556 df-sgrp 17598 df-mnd 17609 df-submnd 17650 df-grp 17740 df-mulg 17856 df-subg 17903 df-cntz 18061 df-cmn 18509 df-dprd 18709 |
This theorem is referenced by: dprdfeq0 18736 dprdub 18739 dpjrid 18776 |
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