dpjf - Metamath Proof Explorer

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Theorem dpjf 20005

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 2737	. . 3 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
2		eqid 2737	. . 3 ⊢ (LSSum‘𝐺) = (LSSum‘𝐺)
3		eqid 2737	. . 3 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
4		eqid 2737	. . 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 19955	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
8		dpjf.3	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
9	7, 8	ffvelcdmd 7041	. . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
10		difssd 4091	. . . . . 6 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼)
11	5, 6, 10	dprdres 19976	. . . . 5 ⊢ (𝜑 → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ⊆ (𝐺 DProd 𝑆)))
12	11	simpld 494	. . . 4 ⊢ (𝜑 → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))
13		dprdsubg 19972	. . . 4 ⊢ (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑋})) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺))
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))) ∈ (SubGrp‘𝐺))
15	5, 6, 8, 3	dpjdisj 20001	. . 3 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0_g‘𝐺)})
16	5, 6, 8, 4	dpjcntz 20000	. . 3 ⊢ (𝜑 → (𝑆‘𝑋) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
17		eqid 2737	. . 3 ⊢ (proj₁‘𝐺) = (proj₁‘𝐺)
18	1, 2, 3, 4, 9, 14, 15, 16, 17	pj1f 19643	. 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 20004	. . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
21	5, 6, 8, 2	dpjlsm 20002	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
22	20, 21	feq12d 6660	. 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 257	1 ⊢ (𝜑 → (𝑃‘𝑋):(𝐺 DProd 𝑆)⟶(𝑆‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∖ cdif 3900 ⊆ wss 3903 {csn 4582 class class class wbr 5100 dom cdm 5634 ↾ cres 5636 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 +_gcplusg 17191 0_gc0g 17373 SubGrpcsubg 19067 Cntzccntz 19261 LSSumclsm 19580 proj₁cpj1 19581 DProd cdprd 19941 dProjcdpj 19942

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-tpos 8180 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-2 12222 df-n0 12416 df-z 12503 df-uz 12766 df-fz 13438 df-fzo 13585 df-seq 13939 df-hash 14268 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-0g 17375 df-gsum 17376 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-mhm 18722 df-submnd 18723 df-grp 18883 df-minusg 18884 df-sbg 18885 df-mulg 19015 df-subg 19070 df-ghm 19159 df-gim 19205 df-cntz 19263 df-oppg 19292 df-lsm 19582 df-pj1 19583 df-cmn 19728 df-dprd 19943 df-dpj 19944

This theorem is referenced by: dpjidcl 20006 dpjghm2 20012 dchrptlem2 27249

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