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Theorem dpjlid 20105

Description: The 𝑋-th index projection acts as the identity on elements of the 𝑋-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
dpjlid	⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴)

**Proof of Theorem dpjlid**
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 2740	. . . 4 ⊢ (proj₁‘𝐺) = (proj₁‘𝐺)
5		dpjlid.3	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
6	1, 2, 3, 4, 5	dpjval 20100	. . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
7	6	fveq1d 6922	. 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 2740	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
10		eqid 2740	. . . 4 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
11		eqid 2740	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
12		eqid 2740	. . . 4 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
13	1, 2	dprdf2 20051	. . . . 5 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
14	13, 5	ffvelcdmd 7119	. . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
15		difssd 4160	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼)
16	1, 2, 15	dprdres 20072	. . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆)))
17	16	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))
18		dprdsubg 20068	. . . . 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 20097	. . . 4 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0_g‘𝐺)})
21	1, 2, 5, 12	dpjcntz 20096	. . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
22	9, 10, 11, 12, 14, 19, 20, 21, 4	pj1lid 19743	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑆‘𝑋)) → (((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴)
23	8, 22	mpdan 686	. 2 ⊢ (𝜑 → (((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴)
24	7, 23	eqtrd 2780	1 ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ∖ cdif 3973 ⊆ wss 3976 {csn 4648 class class class wbr 5166 dom cdm 5700 ↾ cres 5702 ‘cfv 6573 (class class class)co 7448 +_gcplusg 17311 0_gc0g 17499 SubGrpcsubg 19160 Cntzccntz 19355 LSSumclsm 19676 proj₁cpj1 19677 DProd cdprd 20037 dProjcdpj 20038

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-0g 17501 df-gsum 17502 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-gim 19299 df-cntz 19357 df-oppg 19386 df-lsm 19678 df-pj1 19679 df-cmn 19824 df-dprd 20039 df-dpj 20040

This theorem is referenced by: (None)

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