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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 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → 𝐴 ∈ (𝑆‘𝑋)) |
7 | simpr 487 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → 𝑛 = 𝑋) | |
8 | 7 | fveq2d 6669 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → (𝑆‘𝑛) = (𝑆‘𝑋)) |
9 | 6, 8 | eleqtrrd 2916 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ 𝑛 = 𝑋) → 𝐴 ∈ (𝑆‘𝑛)) |
10 | 3, 4 | dprdf2 19123 | . . . . . . . 8 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
11 | 10 | ffvelrnda 6846 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → (𝑆‘𝑛) ∈ (SubGrp‘𝐺)) |
12 | eldprdi.0 | . . . . . . . 8 ⊢ 0 = (0g‘𝐺) | |
13 | 12 | subg0cl 18281 | . . . . . . 7 ⊢ ((𝑆‘𝑛) ∈ (SubGrp‘𝐺) → 0 ∈ (𝑆‘𝑛)) |
14 | 11, 13 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → 0 ∈ (𝑆‘𝑛)) |
15 | 14 | adantr 483 | . . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐼) ∧ ¬ 𝑛 = 𝑋) → 0 ∈ (𝑆‘𝑛)) |
16 | 9, 15 | ifclda 4501 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐼) → if(𝑛 = 𝑋, 𝐴, 0 ) ∈ (𝑆‘𝑛)) |
17 | 3, 4 | dprddomcld 19117 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ V) |
18 | 12 | fvexi 6679 | . . . . . 6 ⊢ 0 ∈ V |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ V) |
20 | eqid 2821 | . . . . 5 ⊢ (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) = (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) | |
21 | 17, 19, 20 | sniffsupp 8867 | . . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) finSupp 0 ) |
22 | 2, 3, 4, 16, 21 | dprdwd 19127 | . . 3 ⊢ (𝜑 → (𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) ∈ 𝑊) |
23 | 1, 22 | eqeltrid 2917 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
24 | eqid 2821 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
25 | dprdgrp 19121 | . . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) | |
26 | grpmnd 18104 | . . . . 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 19128 | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶(Base‘𝐺)) |
30 | 1 | oveq1i 7160 | . . . . 5 ⊢ (𝐹 supp 0 ) = ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) |
31 | eldifsni 4716 | . . . . . . . 8 ⊢ (𝑛 ∈ (𝐼 ∖ {𝑋}) → 𝑛 ≠ 𝑋) | |
32 | 31 | adantl 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐼 ∖ {𝑋})) → 𝑛 ≠ 𝑋) |
33 | ifnefalse 4479 | . . . . . . 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 7858 | . . . . 5 ⊢ (𝜑 → ((𝑛 ∈ 𝐼 ↦ if(𝑛 = 𝑋, 𝐴, 0 )) supp 0 ) ⊆ {𝑋}) |
36 | 30, 35 | eqsstrid 4015 | . . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ {𝑋}) |
37 | 24, 12, 27, 17, 28, 29, 36 | gsumpt 19076 | . . 3 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐹‘𝑋)) |
38 | iftrue 4473 | . . . 4 ⊢ (𝑛 = 𝑋 → if(𝑛 = 𝑋, 𝐴, 0 ) = 𝐴) | |
39 | 1, 38, 28, 5 | fvmptd3 6786 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) = 𝐴) |
40 | 37, 39 | eqtrd 2856 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 𝐴) |
41 | 23, 40 | jca 514 | 1 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ∧ (𝐺 Σg 𝐹) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {crab 3142 Vcvv 3495 ∖ cdif 3933 ifcif 4467 {csn 4561 class class class wbr 5059 ↦ cmpt 5139 dom cdm 5550 ‘cfv 6350 (class class class)co 7150 supp csupp 7824 Xcixp 8455 finSupp cfsupp 8827 Basecbs 16477 0gc0g 16707 Σg cgsu 16708 Mndcmnd 17905 Grpcgrp 18097 SubGrpcsubg 18267 DProd cdprd 19109 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-mulg 18219 df-subg 18270 df-cntz 18441 df-cmn 18902 df-dprd 19111 |
This theorem is referenced by: dprdfeq0 19138 dprdub 19141 dpjrid 19178 |
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