pj1f - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pj1f		Structured version Visualization version GIF version

Theorem pj1f 18468

Description: The left projection function maps a direct subspace sum onto the left factor. (Contributed by Mario Carneiro, 15-Oct-2015.)

**Hypotheses**
Ref	Expression
pj1eu.a	⊢ + = (+_g‘𝐺)
pj1eu.s	⊢ ⊕ = (LSSum‘𝐺)
pj1eu.o	⊢ 0 = (0_g‘𝐺)
pj1eu.z	⊢ 𝑍 = (Cntz‘𝐺)
pj1eu.2	⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
pj1eu.3	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
pj1eu.4	⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
pj1eu.5	⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))
pj1f.p	⊢ 𝑃 = (proj₁‘𝐺)

**Assertion**
Ref	Expression
pj1f	⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)

Proof of Theorem pj1f Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pj1eu.a	. . . . 5 ⊢ + = (+_g‘𝐺)
2		pj1eu.s	. . . . 5 ⊢ ⊕ = (LSSum‘𝐺)
3		pj1eu.o	. . . . 5 ⊢ 0 = (0_g‘𝐺)
4		pj1eu.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
5		pj1eu.2	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
6		pj1eu.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
7		pj1eu.4	. . . . 5 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
8		pj1eu.5	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))
9	1, 2, 3, 4, 5, 6, 7, 8	pj1eu 18467	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))
10		riotacl 6885	. . . 4 ⊢ (∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦) → (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)) ∈ 𝑇)
11	9, 10	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑇 ⊕ 𝑈)) → (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)) ∈ 𝑇)
12	11	fmpttd 6639	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))):(𝑇 ⊕ 𝑈)⟶𝑇)
13		subgrcl 17957	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
14	5, 13	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
15		eqid 2825	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
16	15	subgss 17953	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
17	5, 16	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺))
18	15	subgss 17953	. . . . 5 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
19	6, 18	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐺))
20		pj1f.p	. . . . 5 ⊢ 𝑃 = (proj₁‘𝐺)
21	15, 1, 2, 20	pj1fval 18465	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))
22	14, 17, 19, 21	syl3anc 1494	. . 3 ⊢ (𝜑 → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))
23	22	feq1d 6267	. 2 ⊢ (𝜑 → ((𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇 ↔ (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))):(𝑇 ⊕ 𝑈)⟶𝑇))
24	12, 23	mpbird 249	1 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 ∃!wreu 3119 ∩ cin 3797 ⊆ wss 3798 {csn 4399 ↦ cmpt 4954 ⟶wf 6123 ‘cfv 6127 ℩crio 6870 (class class class)co 6910 Basecbs 16229 +_gcplusg 16312 0_gc0g 16460 Grpcgrp 17783 SubGrpcsubg 17946 Cntzccntz 18105 LSSumclsm 18407 proj₁cpj1 18408

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-cntz 18107 df-lsm 18409 df-pj1 18410

This theorem is referenced by: pj2f 18469 pj1id 18470 pj1eq 18471 pj1ghm 18474 pj1ghm2 18475 lsmhash 18476 dpjf 18817 pj1lmhm 19466 pj1lmhm2 19467 pjdm2 20425 pjf2 20428

W3C validator