![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dpjlid | Structured version Visualization version GIF version |
Description: The 𝑋-th index projection acts as the identity on elements of 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) |
dpjlid.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
dpjlid.4 | ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋)) |
Ref | Expression |
---|---|
dpjlid | ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpjfval.1 | . . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | dpjfval.2 | . . . 4 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
3 | dpjfval.p | . . . 4 ⊢ 𝑃 = (𝐺dProj𝑆) | |
4 | eqid 2778 | . . . 4 ⊢ (proj1‘𝐺) = (proj1‘𝐺) | |
5 | dpjlid.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
6 | 1, 2, 3, 4, 5 | dpjval 18842 | . . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
7 | 6 | fveq1d 6448 | . 2 ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = (((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴)) |
8 | dpjlid.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑆‘𝑋)) | |
9 | eqid 2778 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
10 | eqid 2778 | . . . 4 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺) | |
11 | eqid 2778 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
12 | eqid 2778 | . . . 4 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺) | |
13 | 1, 2 | dprdf2 18793 | . . . . 5 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
14 | 13, 5 | ffvelrnd 6624 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺)) |
15 | difssd 3961 | . . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼) | |
16 | 1, 2, 15 | dprdres 18814 | . . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆))) |
17 | 16 | simpld 490 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) |
18 | dprdsubg 18810 | . . . . 5 ⊢ (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺)) |
20 | 1, 2, 5, 11 | dpjdisj 18839 | . . . 4 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0g‘𝐺)}) |
21 | 1, 2, 5, 12 | dpjcntz 18838 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))) |
22 | 9, 10, 11, 12, 14, 19, 20, 21, 4 | pj1lid 18498 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑆‘𝑋)) → (((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴) |
23 | 8, 22 | mpdan 677 | . 2 ⊢ (𝜑 → (((𝑆‘𝑋)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴) |
24 | 7, 23 | eqtrd 2814 | 1 ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∖ cdif 3789 ⊆ wss 3792 {csn 4398 class class class wbr 4886 dom cdm 5355 ↾ cres 5357 ‘cfv 6135 (class class class)co 6922 +gcplusg 16338 0gc0g 16486 SubGrpcsubg 17972 Cntzccntz 18131 LSSumclsm 18433 proj1cpj1 18434 DProd cdprd 18779 dProjcdpj 18780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-0g 16488 df-gsum 16489 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-ghm 18042 df-gim 18085 df-cntz 18133 df-oppg 18159 df-lsm 18435 df-pj1 18436 df-cmn 18581 df-dprd 18781 df-dpj 18782 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |