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Theorem dpjlid 19993

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 2729	. . . 4 ⊢ (proj₁‘𝐺) = (proj₁‘𝐺)
5		dpjlid.3	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
6	1, 2, 3, 4, 5	dpjval 19988	. . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
7	6	fveq1d 6860	. 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 2729	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
10		eqid 2729	. . . 4 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
11		eqid 2729	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
12		eqid 2729	. . . 4 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
13	1, 2	dprdf2 19939	. . . . 5 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
14	13, 5	ffvelcdmd 7057	. . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
15		difssd 4100	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼)
16	1, 2, 15	dprdres 19960	. . . . . 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 19956	. . . . 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 19985	. . . 4 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0_g‘𝐺)})
21	1, 2, 5, 12	dpjcntz 19984	. . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
22	9, 10, 11, 12, 14, 19, 20, 21, 4	pj1lid 19631	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑆‘𝑋)) → (((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴)
23	8, 22	mpdan 687	. 2 ⊢ (𝜑 → (((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴)
24	7, 23	eqtrd 2764	1 ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ∖ cdif 3911 ⊆ wss 3914 {csn 4589 class class class wbr 5107 dom cdm 5638 ↾ cres 5640 ‘cfv 6511 (class class class)co 7387 +_gcplusg 17220 0_gc0g 17402 SubGrpcsubg 19052 Cntzccntz 19247 LSSumclsm 19564 proj₁cpj1 19565 DProd cdprd 19925 dProjcdpj 19926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-gsum 17405 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-ghm 19145 df-gim 19191 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-pj1 19567 df-cmn 19712 df-dprd 19927 df-dpj 19928

This theorem is referenced by: (None)

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