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Theorem dpjlid 19972

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 2732	. . . 4 ⊢ (proj₁‘𝐺) = (proj₁‘𝐺)
5		dpjlid.3	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
6	1, 2, 3, 4, 5	dpjval 19967	. . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
7	6	fveq1d 6893	. 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 2732	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
10		eqid 2732	. . . 4 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
11		eqid 2732	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
12		eqid 2732	. . . 4 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
13	1, 2	dprdf2 19918	. . . . 5 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
14	13, 5	ffvelcdmd 7087	. . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
15		difssd 4132	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼)
16	1, 2, 15	dprdres 19939	. . . . . 6 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆)))
17	16	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))
18		dprdsubg 19935	. . . . 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 19964	. . . 4 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0_g‘𝐺)})
21	1, 2, 5, 12	dpjcntz 19963	. . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
22	9, 10, 11, 12, 14, 19, 20, 21, 4	pj1lid 19610	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑆‘𝑋)) → (((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴)
23	8, 22	mpdan 685	. 2 ⊢ (𝜑 → (((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴)
24	7, 23	eqtrd 2772	1 ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 class class class wbr 5148 dom cdm 5676 ↾ cres 5678 ‘cfv 6543 (class class class)co 7411 +_gcplusg 17201 0_gc0g 17389 SubGrpcsubg 19036 Cntzccntz 19220 LSSumclsm 19543 proj₁cpj1 19544 DProd cdprd 19904 dProjcdpj 19905

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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-0g 17391 df-gsum 17392 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-ghm 19128 df-gim 19173 df-cntz 19222 df-oppg 19251 df-lsm 19545 df-pj1 19546 df-cmn 19691 df-dprd 19906 df-dpj 19907

This theorem is referenced by: (None)

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