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Theorem dpjlid 19664

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 2738	. . . 4 ⊢ (proj₁‘𝐺) = (proj₁‘𝐺)
5		dpjlid.3	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
6	1, 2, 3, 4, 5	dpjval 19659	. . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
7	6	fveq1d 6776	. 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 2738	. . . 4 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
10		eqid 2738	. . . 4 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
11		eqid 2738	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
12		eqid 2738	. . . 4 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
13	1, 2	dprdf2 19610	. . . . 5 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
14	13, 5	ffvelrnd 6962	. . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
15		difssd 4067	. . . . . . 7 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼)
16	1, 2, 15	dprdres 19631	. . . . . 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 19627	. . . . 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 19656	. . . 4 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0_g‘𝐺)})
21	1, 2, 5, 12	dpjcntz 19655	. . . 4 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
22	9, 10, 11, 12, 14, 19, 20, 21, 4	pj1lid 19307	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑆‘𝑋)) → (((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴)
23	8, 22	mpdan 684	. 2 ⊢ (𝜑 → (((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))‘𝐴) = 𝐴)
24	7, 23	eqtrd 2778	1 ⊢ (𝜑 → ((𝑃‘𝑋)‘𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∖ cdif 3884 ⊆ wss 3887 {csn 4561 class class class wbr 5074 dom cdm 5589 ↾ cres 5591 ‘cfv 6433 (class class class)co 7275 +_gcplusg 16962 0_gc0g 17150 SubGrpcsubg 18749 Cntzccntz 18921 LSSumclsm 19239 proj₁cpj1 19240 DProd cdprd 19596 dProjcdpj 19597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-0g 17152 df-gsum 17153 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-gim 18875 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-pj1 19242 df-cmn 19388 df-dprd 19598 df-dpj 19599

This theorem is referenced by: (None)

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