Theorem pj1id 19792

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 19220	. . . . . . 7 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		eqid 2765	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 19216	. . . . . . 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 19216	. . . . . . 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 19788	. . . . 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 592	. . . 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 19789	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦))
21		riotacl2 7389	. . . . 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 2865	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)})
24		oveq1 7423	. . . . . . 7 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑥 + 𝑦) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
25	24	eqeq2d 2776	. . . . . 6 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑋 = (𝑥 + 𝑦) ↔ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
26	25	rexbidv 3191	. . . . 5 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
27	26	elrab 3652	. . . 4 ⊢ (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)} ↔ (((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
28	27	simprbi 503	. . 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 785	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
31	3	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝐺 ∈ Grp)
32	9	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑈 ⊆ (Base‘𝐺))
33	6	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (Base‘𝐺))
34		simplr 781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑇 ⊕ 𝑈))
35	12, 17	lsmcom2 19748	. . . . . . . . 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
38	34, 37	eleqtrd 2867	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑈 ⊕ 𝑇))
39	4, 11, 12, 13	pj1val 19788	. . . . . 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 19790	. . . . . . . . 9 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
42	41	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
43	42, 34	ffvelcdmd 7084	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇)
44	19	ad2antrr 739	. . . . . . . . . 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 783	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑦 ∈ 𝑈)
47	11, 17	cntzi 19422	. . . . . . . . 9 ⊢ ((((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍‘𝑈) ∧ 𝑦 ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
48	45, 46, 47	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
49	30, 48	eqtrd 2800	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
50		oveq2 7424	. . . . . . . 8 ⊢ (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑦 + 𝑣) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
51	50	rspceeqv 3606	. . . . . . 7 ⊢ ((((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋))) → ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣))
52	43, 49, 51	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣))
53		simpll 779	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝜑)
54		incom 4162	. . . . . . . . . 10 ⊢ (𝑈 ∩ 𝑇) = (𝑇 ∩ 𝑈)
55	54, 18	eqtrid 2812	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 })
56	17, 1, 7, 19	cntzrecd 19771	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇))
57	11, 12, 16, 17, 7, 1, 55, 56	pj1eu 19789	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑈 ⊕ 𝑇)) → ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣))
58	53, 38, 57	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣))
59		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑢 = 𝑦 → (𝑢 + 𝑣) = (𝑦 + 𝑣))
60	59	eqeq2d 2776	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (𝑋 = (𝑢 + 𝑣) ↔ 𝑋 = (𝑦 + 𝑣)))
61	60	rexbidv 3191	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → (∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣) ↔ ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣)))
62	61	riota2 7398	. . . . . . 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 596	. . . . . 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 2800	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = 𝑦)
66	65	oveq2d 7432	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
67	30, 66	eqtr4d 2803	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
68	29, 67	rexlimddv 3174	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  ∃!wreu 3369  {crab 3418   ∩ cin 3905   ⊆ wss 3906  {csn 4591  ⟶wf 6536  ‘cfv 6540  ℩crio 7372  (class class class)co 7416  Basecbs 17286  +_gcplusg 17327  0_gc0g 17509  Grpcgrp 19023  SubGrpcsubg 19209  Cntzccntz 19408  LSSumclsm 19727  proj₁cpj1 19728

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  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 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-sets 17241  df-slot 17259  df-ndx 17271  df-base 17287  df-ress 17308  df-plusg 17340  df-0g 17511  df-mgm 18715  df-sgrp 18798  df-mnd 18814  df-grp 19026  df-minusg 19027  df-sbg 19028  df-subg 19212  df-cntz 19410  df-lsm 19729  df-pj1 19730

This theorem is used by:  pj1eq  19793  pj1ghm  19796  pj1lmhm  21250