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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 19348 | . . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
8 | dpjf.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
9 | 7, 8 | ffvelrnd 6883 | . . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺)) |
10 | difssd 4033 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼) | |
11 | 5, 6, 10 | dprdres 19369 | . . . . 5 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆))) |
12 | 11 | simpld 498 | . . . 4 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) |
13 | dprdsubg 19365 | . . . 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 19394 | . . 3 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0g‘𝐺)}) |
16 | 5, 6, 8, 4 | dpjcntz 19393 | . . 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 19041 | . 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 19397 | . . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
21 | 5, 6, 8, 2 | dpjlsm 19395 | . . 3 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
22 | 20, 21 | feq12d 6511 | . 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 260 | 1 ⊢ (𝜑 → (𝑃‘𝑋):(𝐺 DProd 𝑆)⟶(𝑆‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∖ cdif 3850 ⊆ wss 3853 {csn 4527 class class class wbr 5039 dom cdm 5536 ↾ cres 5538 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 +gcplusg 16749 0gc0g 16898 SubGrpcsubg 18491 Cntzccntz 18663 LSSumclsm 18977 proj1cpj1 18978 DProd cdprd 19334 dProjcdpj 19335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-0g 16900 df-gsum 16901 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-mulg 18443 df-subg 18494 df-ghm 18574 df-gim 18617 df-cntz 18665 df-oppg 18692 df-lsm 18979 df-pj1 18980 df-cmn 19126 df-dprd 19336 df-dpj 19337 |
This theorem is referenced by: dpjidcl 19399 dpjghm2 19405 dchrptlem2 26100 |
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