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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 723 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → 𝐴 ∈ (𝑆‘𝑋)) |
7 | simpr 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → 𝑛 = 𝑋) | |
8 | 7 | fveq2d 6888 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → (𝑆‘𝑛) = (𝑆‘𝑋)) |
9 | 6, 8 | eleqtrrd 2830 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → 𝐴 ∈ (𝑆‘𝑛)) |
10 | 3, 4 | dprdf2 19926 | . . . . . . . 8 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
11 | 10 | ffvelcdmda 7079 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → (𝑆‘𝑛) ∈ (SubGrp‘𝐺)) |
12 | eldprdi.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝐺) | |
13 | 12 | subg0cl 19058 | . . . . . . 7 ⊢ ((𝑆‘𝑛) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆‘𝑛)) |
14 | 11, 13 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 0 ∈ (𝑆‘𝑛)) |
15 | 14 | adantr 480 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ ¬ 𝑛 = 𝑋) → 0 ∈ (𝑆‘𝑛)) |
16 | 9, 15 | ifclda 4558 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → if(𝑛 = 𝑋, 𝐴, 0 ) ∈ (𝑆‘𝑛)) |
17 | 3, 4 | dprddomcld 19920 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
18 | 12 | fvexi 6898 | . . . . . 6 ⊢ 0 ∈ V |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ V) |
20 | eqid 2726 | . . . . 5 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) | |
21 | 17, 19, 20 | sniffsupp 9394 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) finSupp 0 ) |
22 | 2, 3, 4, 16, 21 | dprdwd 19930 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) ∈ 𝑊) |
23 | 1, 22 | eqeltrid 2831 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
24 | eqid 2726 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
25 | dprdgrp 19924 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
26 | grpmnd 18867 | . . . . 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 19931 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
30 | 1 | oveq1i 7414 | . . . . 5 ⊢ (𝐹 supp 0 ) = ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) |
31 | eldifsni 4788 | . . . . . . . 8 ⊢ (𝑛 ∈ (𝐼 ∖ {𝑋}) → 𝑛 ≠ 𝑋) | |
32 | 31 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐼 ∖ {𝑋})) → 𝑛 ≠ 𝑋) |
33 | ifnefalse 4535 | . . . . . . 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 8183 | . . . . 5 ⊢ (𝜑 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) ⊆ {𝑋}) |
36 | 30, 35 | eqsstrid 4025 | . . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋}) |
37 | 24, 12, 27, 17, 28, 29, 36 | gsumpt 19879 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐹‘𝑋)) |
38 | iftrue 4529 | . . . 4 ⊢ (𝑛 = 𝑋 → if(𝑛 = 𝑋, 𝐴, 0 ) = 𝐴) | |
39 | 1, 38, 28, 5 | fvmptd3 7014 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) = 𝐴) |
40 | 37, 39 | eqtrd 2766 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 𝐴) |
41 | 23, 40 | jca 511 | 1 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ∧ (𝐺 Σg 𝐹) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 {crab 3426 Vcvv 3468 ∖ cdif 3940 ifcif 4523 {csn 4623 class class class wbr 5141 ↦ cmpt 5224 dom cdm 5669 ‘cfv 6536 (class class class)co 7404 supp csupp 8143 Xcixp 8890 finSupp cfsupp 9360 Basecbs 17150 0gc0g 17391 Σg cgsu 17392 Mndcmnd 18664 Grpcgrp 18860 SubGrpcsubg 19044 DProd cdprd 19912 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14293 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-0g 17393 df-gsum 17394 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-submnd 18711 df-grp 18863 df-mulg 18993 df-subg 19047 df-cntz 19230 df-cmn 19699 df-dprd 19914 |
This theorem is referenced by: dprdfeq0 19941 dprdub 19944 dpjrid 19981 |
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