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Mirrors > Home > MPE Home > Th. List > dpjf | Structured version Visualization version GIF version |
Description: The 𝑋-th index projection is a function from the direct product to the 𝑋-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.) |
Ref | Expression |
---|---|
dpjfval.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dpjfval.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dpjfval.p | ⊢ 𝑃 = (𝐺dProj𝑆) |
dpjf.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
dpjf | ⊢ (𝜑 → (𝑃‘𝑋):(𝐺 DProd 𝑆)⟶(𝑆‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
2 | eqid 2736 | . . 3 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺) | |
3 | eqid 2736 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | eqid 2736 | . . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
5 | dpjfval.1 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
6 | dpjfval.2 | . . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
7 | 5, 6 | dprdf2 19777 | . . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
8 | dpjf.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
9 | 7, 8 | ffvelcdmd 7032 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺)) |
10 | difssd 4090 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼) | |
11 | 5, 6, 10 | dprdres 19798 | . . . . 5 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆))) |
12 | 11 | simpld 495 | . . . 4 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) |
13 | dprdsubg 19794 | . . . 4 ⊢ (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) |
15 | 5, 6, 8, 3 | dpjdisj 19823 | . . 3 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0g‘𝐺)}) |
16 | 5, 6, 8, 4 | dpjcntz 19822 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
17 | eqid 2736 | . . 3 ⊢ (proj1‘𝐺) = (proj1‘𝐺) | |
18 | 1, 2, 3, 4, 9, 14, 15, 16, 17 | pj1f 19470 | . 2 ⊢ (𝜑 → ((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))):((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))⟶(𝑆‘𝑋)) |
19 | dpjfval.p | . . . 4 ⊢ 𝑃 = (𝐺dProj𝑆) | |
20 | 5, 6, 19, 17, 8 | dpjval 19826 | . . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
21 | 5, 6, 8, 2 | dpjlsm 19824 | . . 3 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
22 | 20, 21 | feq12d 6653 | . 2 ⊢ (𝜑 → ((𝑃‘𝑋):(𝐺 DProd 𝑆)⟶(𝑆‘𝑋) ↔ ((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))):((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))⟶(𝑆‘𝑋))) |
23 | 18, 22 | mpbird 256 | 1 ⊢ (𝜑 → (𝑃‘𝑋):(𝐺 DProd 𝑆)⟶(𝑆‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∖ cdif 3905 ⊆ wss 3908 {csn 4584 class class class wbr 5103 dom cdm 5631 ↾ cres 5633 ⟶wf 6489 ‘cfv 6493 (class class class)co 7353 +gcplusg 17125 0gc0g 17313 SubGrpcsubg 18913 Cntzccntz 19086 LSSumclsm 19407 proj1cpj1 19408 DProd cdprd 19763 dProjcdpj 19764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-tpos 8153 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-map 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-fzo 13560 df-seq 13899 df-hash 14223 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-0g 17315 df-gsum 17316 df-mre 17458 df-mrc 17459 df-acs 17461 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-mhm 18593 df-submnd 18594 df-grp 18743 df-minusg 18744 df-sbg 18745 df-mulg 18864 df-subg 18916 df-ghm 18997 df-gim 19040 df-cntz 19088 df-oppg 19115 df-lsm 19409 df-pj1 19410 df-cmn 19555 df-dprd 19765 df-dpj 19766 |
This theorem is referenced by: dpjidcl 19828 dpjghm2 19834 dchrptlem2 26597 |
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