Theorem pj1id 19770

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 19198	. . . . . . 7 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 19194	. . . . . . 7 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
6	1, 5	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺))
7		pj1eu.3	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
8	4	subgss 19194	. . . . . . 7 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
9	7, 8	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐺))
10	3, 6, 9	3jca 1146	. . . . 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 19766	. . . . 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 591	. . . 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 19767	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦))
21		riotacl2 7385	. . . . 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 18	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)})
23	15, 22	eqeltrd 2863	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)})
24		oveq1 7419	. . . . . . 7 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑥 + 𝑦) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
25	24	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑋 = (𝑥 + 𝑦) ↔ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
26	25	rexbidv 3189	. . . . 5 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
27	26	elrab 3651	. . . 4 ⊢ (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)} ↔ (((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
28	27	simprbi 502	. . 3 ⊢ (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)} → ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
29	23, 28	syl 18	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
30		simprr 784	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
31	3	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝐺 ∈ Grp)
32	9	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑈 ⊆ (Base‘𝐺))
33	6	ad2antrr 738	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (Base‘𝐺))
34		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑇 ⊕ 𝑈))
35	12, 17	lsmcom2 19726	. . . . . . . . 9 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
36	1, 7, 19, 35	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
37	36	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
38	34, 37	eleqtrd 2865	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑈 ⊕ 𝑇))
39	4, 11, 12, 13	pj1val 19766	. . . . . 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 1400	. . . . 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 19768	. . . . . . . . 9 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
42	41	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
43	42, 34	ffvelcdmd 7082	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇)
44	19	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (𝑍‘𝑈))
45	44, 43	sseldd 3939	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍‘𝑈))
46		simprl 782	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑦 ∈ 𝑈)
47	11, 17	cntzi 19400	. . . . . . . . 9 ⊢ ((((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍‘𝑈) ∧ 𝑦 ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
48	45, 46, 47	syl2anc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
49	30, 48	eqtrd 2798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
50		oveq2 7420	. . . . . . . 8 ⊢ (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑦 + 𝑣) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
51	50	rspceeqv 3605	. . . . . . 7 ⊢ ((((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋))) → ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣))
52	43, 49, 51	syl2anc 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣))
53		simpll 778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝜑)
54		incom 4163	. . . . . . . . . 10 ⊢ (𝑈 ∩ 𝑇) = (𝑇 ∩ 𝑈)
55	54, 18	eqtrid 2810	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 })
56	17, 1, 7, 19	cntzrecd 19749	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇))
57	11, 12, 16, 17, 7, 1, 55, 56	pj1eu 19767	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑈 ⊕ 𝑇)) → ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣))
58	53, 38, 57	syl2anc 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣))
59		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑢 = 𝑦 → (𝑢 + 𝑣) = (𝑦 + 𝑣))
60	59	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (𝑋 = (𝑢 + 𝑣) ↔ 𝑋 = (𝑦 + 𝑣)))
61	60	rexbidv 3189	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → (∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣) ↔ ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣)))
62	61	riota2 7394	. . . . . . 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 595	. . . . . 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 235	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (℩𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦)
65	40, 64	eqtrd 2798	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = 𝑦)
66	65	oveq2d 7428	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
67	30, 66	eqtr4d 2801	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
68	29, 67	rexlimddv 3172	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  ∃!wreu 3367  {crab 3416   ∩ cin 3905   ⊆ wss 3906  {csn 4590  ⟶wf 6534  ‘cfv 6538  ℩crio 7368  (class class class)co 7412  Basecbs 17270  +_gcplusg 17311  0_gc0g 17493  Grpcgrp 19001  SubGrpcsubg 19187  Cntzccntz 19386  LSSumclsm 19705  proj₁cpj1 19706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-grp 19004  df-minusg 19005  df-sbg 19006  df-subg 19190  df-cntz 19388  df-lsm 19707  df-pj1 19708

This theorem is referenced by:  pj1eq  19771  pj1ghm  19774  pj1lmhm  21202