Theorem pj1id 19732

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 19162	. . . . . . 7 ⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	4	subgss 19158	. . . . . . 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 19158	. . . . . . 7 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐺))
10	3, 6, 9	3jca 1127	. . . . 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 19728	. . . . 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 580	. . . 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 19729	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦))
21		riotacl2 7404	. . . . 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 2839	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)})
24		oveq1 7438	. . . . . . 7 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑥 + 𝑦) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
25	24	eqeq2d 2746	. . . . . 6 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑋 = (𝑥 + 𝑦) ↔ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
26	25	rexbidv 3177	. . . . 5 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
27	26	elrab 3695	. . . 4 ⊢ (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)} ↔ (((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
28	27	simprbi 496	. . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝐺 ∈ Grp)
32	9	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑈 ⊆ (Base‘𝐺))
33	6	ad2antrr 726	. . . . . 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 19688	. . . . . . . . 9 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
36	1, 7, 19, 35	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
37	36	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
38	34, 37	eleqtrd 2841	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑈 ⊕ 𝑇))
39	4, 11, 12, 13	pj1val 19728	. . . . . 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 1372	. . . . 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 19730	. . . . . . . . 9 ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
42	41	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
43	42, 34	ffvelcdmd 7105	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇)
44	19	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (𝑍‘𝑈))
45	44, 43	sseldd 3996	. . . . . . . . 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 19360	. . . . . . . . 9 ⊢ ((((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍‘𝑈) ∧ 𝑦 ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
48	45, 46, 47	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
49	30, 48	eqtrd 2775	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
50		oveq2 7439	. . . . . . . 8 ⊢ (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑦 + 𝑣) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
51	50	rspceeqv 3645	. . . . . . 7 ⊢ ((((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋))) → ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣))
52	43, 49, 51	syl2anc 584	. . . . . 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 4217	. . . . . . . . . 10 ⊢ (𝑈 ∩ 𝑇) = (𝑇 ∩ 𝑈)
55	54, 18	eqtrid 2787	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 })
56	17, 1, 7, 19	cntzrecd 19711	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇))
57	11, 12, 16, 17, 7, 1, 55, 56	pj1eu 19729	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑈 ⊕ 𝑇)) → ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣))
58	53, 38, 57	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣))
59		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑢 = 𝑦 → (𝑢 + 𝑣) = (𝑦 + 𝑣))
60	59	eqeq2d 2746	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (𝑋 = (𝑢 + 𝑣) ↔ 𝑋 = (𝑦 + 𝑣)))
61	60	rexbidv 3177	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → (∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣) ↔ ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣)))
62	61	riota2 7413	. . . . . . 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 584	. . . . . 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 2775	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = 𝑦)
66	65	oveq2d 7447	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
67	30, 66	eqtr4d 2778	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
68	29, 67	rexlimddv 3159	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  ∃!wreu 3376  {crab 3433   ∩ cin 3962   ⊆ wss 3963  {csn 4631  ⟶wf 6559  ‘cfv 6563  ℩crio 7387  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  0_gc0g 17486  Grpcgrp 18964  SubGrpcsubg 19151  Cntzccntz 19346  LSSumclsm 19667  proj₁cpj1 19668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cntz 19348  df-lsm 19669  df-pj1 19670

This theorem is referenced by:  pj1eq  19733  pj1ghm  19736  pj1lmhm  21117