dpjf - Metamath Proof Explorer

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Theorem dpjf 20120

Description: The 𝑋-th index projection is a function from the direct product to 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)
dpjf.3	⊢ (𝜑 → 𝑋 ∈ 𝐼)

**Assertion**
Ref	Expression
dpjf	⊢ (𝜑 → (𝑃‘𝑋):(𝐺 DProd 𝑆)⟶(𝑆‘𝑋))

**Proof of Theorem dpjf**
Step	Hyp	Ref	Expression
1		eqid 2765	. . 3 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
2		eqid 2765	. . 3 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
3		eqid 2765	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
4		eqid 2765	. . 3 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
5		dpjfval.1	. . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
6		dpjfval.2	. . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼)
7	5, 6	dprdf2 20070	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
8		dpjf.3	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
9	7, 8	ffvelcdmd 7070	. . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
10		difssd 4093	. . . . . 6 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼)
11	5, 6, 10	dprdres 20091	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆)))
12	11	simpld 499	. . . 4 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))
13		dprdsubg 20087	. . . 4 ⊢ (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺))
14	12, 13	syl 18	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺))
15	5, 6, 8, 3	dpjdisj 20116	. . 3 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0_g‘𝐺)})
16	5, 6, 8, 4	dpjcntz 20115	. . 3 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
17		eqid 2765	. . 3 ⊢ (proj₁‘𝐺) = (proj₁‘𝐺)
18	1, 2, 3, 4, 9, 14, 15, 16, 17	pj1f 19758	. 2 ⊢ (𝜑 → ((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))):((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))⟶(𝑆‘𝑋))
19		dpjfval.p	. . . 4 ⊢ 𝑃 = (𝐺dProj𝑆)
20	5, 6, 19, 17, 8	dpjval 20119	. . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
21	5, 6, 8, 2	dpjlsm 20117	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
22	20, 21	feq12d 6683	. 2 ⊢ (𝜑 → ((𝑃‘𝑋):(𝐺 DProd 𝑆)⟶(𝑆‘𝑋) ↔ ((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))):((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))⟶(𝑆‘𝑋)))
23	18, 22	mpbird 260	1 ⊢ (𝜑 → (𝑃‘𝑋):(𝐺 DProd 𝑆)⟶(𝑆‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1563 ∈ wcel 2145 ∖ cdif 3904 ⊆ wss 3907 {csn 4585 class class class wbr 5105 dom cdm 5652 ↾ cres 5654 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 +_gcplusg 17300 0_gc0g 17482 SubGrpcsubg 19177 Cntzccntz 19376 LSSumclsm 19695 proj₁cpj1 19696 DProd cdprd 20056 dProjcdpj 20057

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-tpos 8210 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-fzo 13674 df-seq 14029 df-hash 14358 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-0g 17484 df-gsum 17485 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-mhm 18831 df-submnd 18832 df-grp 18993 df-minusg 18994 df-sbg 18995 df-mulg 19125 df-subg 19180 df-ghm 19275 df-gim 19320 df-cntz 19378 df-oppg 19407 df-lsm 19697 df-pj1 19698 df-cmn 19843 df-dprd 20058 df-dpj 20059

This theorem is referenced by: dpjidcl 20121 dpjghm2 20127 dchrptlem2 27387

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