Theorem pj1id 19665

Description: Any element of a direct subspace sum can be decomposed into projections onto the left and right factors. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
pj1eu.a	⊢ + = (+g‘𝐺)
pj1eu.s	⊢ ⊕ = (LSSum‘𝐺)
pj1eu.o	⊢ 0 = (0g‘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
pj1id	⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))

Proof of Theorem pj1id

Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pj1eu.2	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
2		subgrcl 19098	. . . . . . 7 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 19094	. . . . . . 7 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
6	1, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺))
7		pj1eu.3	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
8	4	subgss 19094	. . . . . . 7 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐺))
10	3, 6, 9	3jca 1129	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)))
11		pj1eu.a	. . . . . 6 ⊢ + = (+g‘𝐺)
12		pj1eu.s	. . . . . 6 ⊢ ⊕ = (LSSum‘𝐺)
13		pj1f.p	. . . . . 6 ⊢ 𝑃 = (proj1‘𝐺)
14	4, 11, 12, 13	pj1val 19661	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)))
15	10, 14	sylan 581	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)))
16		pj1eu.o	. . . . . 6 ⊢ 0 = (0g‘𝐺)
17		pj1eu.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝐺)
18		pj1eu.4	. . . . . 6 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
19		pj1eu.5	. . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))
20	11, 12, 16, 17, 1, 7, 18, 19	pj1eu 19662	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦))
21		riotacl2 7333	. . . . 5 ⊢ (∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦) → (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)})
22	20, 21	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)})
23	15, 22	eqeltrd 2837	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)})
24		oveq1 7367	. . . . . . 7 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑥 + 𝑦) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
25	24	eqeq2d 2748	. . . . . 6 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑋 = (𝑥 + 𝑦) ↔ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
26	25	rexbidv 3162	. . . . 5 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
27	26	elrab 3635	. . . 4 ⊢ (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)} ↔ (((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
28	27	simprbi 497	. . 3 ⊢ (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)} → ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
29	23, 28	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
30		simprr 773	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
31	3	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝐺 ∈ Grp)
32	9	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑈 ⊆ (Base‘𝐺))
33	6	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (Base‘𝐺))
34		simplr 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑇 ⊕ 𝑈))
35	12, 17	lsmcom2 19621	. . . . . . . . 9 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
36	1, 7, 19, 35	syl3anc 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
37	36	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
38	34, 37	eleqtrd 2839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑈 ⊕ 𝑇))
39	4, 11, 12, 13	pj1val 19661	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑈 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ (Base‘𝐺)) ∧ 𝑋 ∈ (𝑈 ⊕ 𝑇)) → ((𝑈𝑃𝑇)‘𝑋) = (℩𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣)))
40	31, 32, 33, 38, 39	syl31anc 1376	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = (℩𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣)))
41	11, 12, 16, 17, 1, 7, 18, 19, 13	pj1f 19663	. . . . . . . . 9 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
42	41	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
43	42, 34	ffvelcdmd 7031	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇)
44	19	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (𝑍‘𝑈))
45	44, 43	sseldd 3923	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍‘𝑈))
46		simprl 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑦 ∈ 𝑈)
47	11, 17	cntzi 19295	. . . . . . . . 9 ⊢ ((((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍‘𝑈) ∧ 𝑦 ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
48	45, 46, 47	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
49	30, 48	eqtrd 2772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
50		oveq2 7368	. . . . . . . 8 ⊢ (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑦 + 𝑣) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
51	50	rspceeqv 3588	. . . . . . 7 ⊢ ((((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋))) → ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣))
52	43, 49, 51	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣))
53		simpll 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝜑)
54		incom 4150	. . . . . . . . . 10 ⊢ (𝑈 ∩ 𝑇) = (𝑇 ∩ 𝑈)
55	54, 18	eqtrid 2784	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 })
56	17, 1, 7, 19	cntzrecd 19644	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇))
57	11, 12, 16, 17, 7, 1, 55, 56	pj1eu 19662	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑈 ⊕ 𝑇)) → ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣))
58	53, 38, 57	syl2anc 585	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣))
59		oveq1 7367	. . . . . . . . . 10 ⊢ (𝑢 = 𝑦 → (𝑢 + 𝑣) = (𝑦 + 𝑣))
60	59	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (𝑋 = (𝑢 + 𝑣) ↔ 𝑋 = (𝑦 + 𝑣)))
61	60	rexbidv 3162	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → (∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣) ↔ ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣)))
62	61	riota2 7342	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑈 ∧ ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣)) → (∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣) ↔ (℩𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦))
63	46, 58, 62	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣) ↔ (℩𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦))
64	52, 63	mpbid 232	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (℩𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦)
65	40, 64	eqtrd 2772	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = 𝑦)
66	65	oveq2d 7376	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
67	30, 66	eqtr4d 2775	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
68	29, 67	rexlimddv 3145	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  ∃!wreu 3341  {crab 3390   ∩ cin 3889   ⊆ wss 3890  {csn 4568  ⟶wf 6488  ‘cfv 6492  ℩crio 7316  (class class class)co 7360  Basecbs 17170  +_gcplusg 17211  0_gc0g 17393  Grpcgrp 18900  SubGrpcsubg 19087  Cntzccntz 19281  LSSumclsm 19600  proj₁cpj1 19601

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-sbg 18905  df-subg 19090  df-cntz 19283  df-lsm 19602  df-pj1 19603

This theorem is referenced by:  pj1eq  19666  pj1ghm  19669  pj1lmhm  21087