dpjf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dpjf		Structured version Visualization version GIF version

Theorem dpjf 20077

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 20027	. . . 4 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))
8		dpjf.3	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
9	7, 8	ffvelcdmd 7105	. . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (SubGrp‘𝐺))
10		difssd 4137	. . . . . 6 ⊢ (𝜑 → (𝐼 ∖ {𝑋}) ⊆ 𝐼)
11	5, 6, 10	dprdres 20048	. . . . 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 20044	. . . 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 20073	. . 3 ⊢ (𝜑 → ((𝑆‘𝑋) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) = {(0_g‘𝐺)})
16	5, 6, 8, 4	dpjcntz 20072	. . 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 19715	. 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 20076	. . 3 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)(proj₁‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
21	5, 6, 8, 2	dpjlsm 20074	. . 3 ⊢ (𝜑 → (𝐺 DProd 𝑆) = ((𝑆‘𝑋)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
22	20, 21	feq12d 6724	. 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 1540 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 +_gcplusg 17297 0_gc0g 17484 SubGrpcsubg 19138 Cntzccntz 19333 LSSumclsm 19652 proj₁cpj1 19653 DProd cdprd 20013 dProjcdpj 20014

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-gsum 17487 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-gim 19277 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-pj1 19655 df-cmn 19800 df-dprd 20015 df-dpj 20016

This theorem is referenced by: dpjidcl 20078 dpjghm2 20084 dchrptlem2 27309

W3C validator