Theorem pj1id 19089

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 18548	. . . . . . 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 18544	. . . . . . 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 18544	. . . . . . 7 ⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐺))
10	3, 6, 9	3jca 1130	. . . . 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 19085	. . . . 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 583	. . . 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 19086	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦))
21		riotacl2 7187	. . . . 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 2838	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)})
24		oveq1 7220	. . . . . . 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 3216	. . . . 5 ⊢ (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
27	26	elrab 3602	. . . 4 ⊢ (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥 ∈ 𝑇 ∣ ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)} ↔ (((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ ∃𝑦 ∈ 𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
28	27	simprbi 500	. . 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 19044	. . . . . . . . 9 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) = (𝑈 ⊕ 𝑇))
36	1, 7, 19, 35	syl3anc 1373	. . . . . . . 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 2840	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑈 ⊕ 𝑇))
39	4, 11, 12, 13	pj1val 19085	. . . . . 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 1375	. . . . 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 19087	. . . . . . . . 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	ffvelrnd 6905	. . . . . . 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 3902	. . . . . . . . 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 18723	. . . . . . . . 9 ⊢ ((((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍‘𝑈) ∧ 𝑦 ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
48	45, 46, 47	syl2anc 587	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
49	30, 48	eqtrd 2777	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
50		oveq2 7221	. . . . . . . 8 ⊢ (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑦 + 𝑣) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
51	50	rspceeqv 3552	. . . . . . 7 ⊢ ((((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋))) → ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣))
52	43, 49, 51	syl2anc 587	. . . . . 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 4115	. . . . . . . . . 10 ⊢ (𝑈 ∩ 𝑇) = (𝑇 ∩ 𝑈)
55	54, 18	syl5eq 2790	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∩ 𝑇) = { 0 })
56	17, 1, 7, 19	cntzrecd 19068	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇))
57	11, 12, 16, 17, 7, 1, 55, 56	pj1eu 19086	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑈 ⊕ 𝑇)) → ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣))
58	53, 38, 57	syl2anc 587	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃!𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣))
59		oveq1 7220	. . . . . . . . . 10 ⊢ (𝑢 = 𝑦 → (𝑢 + 𝑣) = (𝑦 + 𝑣))
60	59	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → (𝑋 = (𝑢 + 𝑣) ↔ 𝑋 = (𝑦 + 𝑣)))
61	60	rexbidv 3216	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → (∃𝑣 ∈ 𝑇 𝑋 = (𝑢 + 𝑣) ↔ ∃𝑣 ∈ 𝑇 𝑋 = (𝑦 + 𝑣)))
62	61	riota2 7196	. . . . . . 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 587	. . . . . 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 2777	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = 𝑦)
66	65	oveq2d 7229	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
67	30, 66	eqtr4d 2780	. 2 ⊢ (((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) ∧ (𝑦 ∈ 𝑈 ∧ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
68	29, 67	rexlimddv 3210	1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∃wrex 3062  ∃!wreu 3063  {crab 3065   ∩ cin 3865   ⊆ wss 3866  {csn 4541  ⟶wf 6376  ‘cfv 6380  ℩crio 7169  (class class class)co 7213  Basecbs 16760  +_gcplusg 16802  0_gc0g 16944  Grpcgrp 18365  SubGrpcsubg 18537  Cntzccntz 18709  LSSumclsm 19023  proj₁cpj1 19024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-minusg 18369  df-sbg 18370  df-subg 18540  df-cntz 18711  df-lsm 19025  df-pj1 19026

This theorem is referenced by:  pj1eq  19090  pj1ghm  19093  pj1lmhm  20137