pj1f - Metamath Proof Explorer

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Theorem pj1f 18317

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 18316	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))
10		riotacl 6768	. . . 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		eqid 2771	. . 3 ⊢ (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)))
13	11, 12	fmptd 6527	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))):(𝑇 ⊕ 𝑈)⟶𝑇)
14		subgrcl 17807	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
15	5, 14	syl 17	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
16		eqid 2771	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
17	16	subgss 17803	. . . . 5 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
18	5, 17	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺))
19	16	subgss 17803	. . . . 5 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
20	6, 19	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐺))
21		pj1f.p	. . . . 5 ⊢ 𝑃 = (proj₁‘𝐺)
22	16, 1, 2, 21	pj1fval 18314	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))
23	15, 18, 20, 22	syl3anc 1476	. . 3 ⊢ (𝜑 → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))
24	23	feq1d 6170	. 2 ⊢ (𝜑 → ((𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇 ↔ (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))):(𝑇 ⊕ 𝑈)⟶𝑇))
25	13, 24	mpbird 247	1 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∃wrex 3062 ∃!wreu 3063 ∩ cin 3722 ⊆ wss 3723 {csn 4316 ↦ cmpt 4863 ⟶wf 6027 ‘cfv 6031 ℩crio 6753 (class class class)co 6793 Basecbs 16064 +_gcplusg 16149 0_gc0g 16308 Grpcgrp 17630 SubGrpcsubg 17796 Cntzccntz 17955 LSSumclsm 18256 proj₁cpj1 18257

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215

This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-minusg 17634 df-sbg 17635 df-subg 17799 df-cntz 17957 df-lsm 18258 df-pj1 18259

This theorem is referenced by: pj2f 18318 pj1id 18319 pj1eq 18320 pj1ghm 18323 pj1ghm2 18324 lsmhash 18325 dpjf 18664 pj1lmhm 19313 pj1lmhm2 19314 pjdm2 20272 pjf2 20275

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