Theorem oppglsm 18497

Description: The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
oppglsm.o	⊢ 𝑂 = (oppg‘𝐺)
oppglsm.p	⊢ ⊕ = (LSSum‘𝐺)

Assertion

Ref	Expression
oppglsm	⊢ (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇)

Proof of Theorem oppglsm

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

Step	Hyp	Ref	Expression
1		eqid 2795	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2795	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		oppglsm.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝐺)
4	1, 2, 3	lsmfval 18493	. . . . . . 7 ⊢ (𝐺 ∈ V → ⊕ = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))))
5	4	tposeqd 7746	. . . . . 6 ⊢ (𝐺 ∈ V → tpos ⊕ = tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))))
6		eqid 2795	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
7	6	reldmmpo 7141	. . . . . . . . . . . 12 ⊢ Rel dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
8	6	mpofun 7132	. . . . . . . . . . . . 13 ⊢ Fun (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
9		funforn 6465	. . . . . . . . . . . . 13 ⊢ (Fun (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) ↔ (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)):dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))–onto→ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)))
10	8, 9	mpbi 231	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)):dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))–onto→ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
11		tposfo2 7766	. . . . . . . . . . . 12 ⊢ (Rel dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) → ((𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)):dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))–onto→ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) → tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)):◡dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))–onto→ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))))
12	7, 10, 11	mp2 9	. . . . . . . . . . 11 ⊢ tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)):◡dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))–onto→ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
13		forn 6461	. . . . . . . . . . 11 ⊢ (tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)):◡dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))–onto→ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) → ran tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)))
14	12, 13	ax-mp 5	. . . . . . . . . 10 ⊢ ran tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
15		oppglsm.o	. . . . . . . . . . . . . . . 16 ⊢ 𝑂 = (oppg‘𝐺)
16		eqid 2795	. . . . . . . . . . . . . . . 16 ⊢ (+g‘𝑂) = (+g‘𝑂)
17	2, 15, 16	oppgplus 18218	. . . . . . . . . . . . . . 15 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑥)
18	17	eqcomi 2804	. . . . . . . . . . . . . 14 ⊢ (𝑦(+g‘𝐺)𝑥) = (𝑥(+g‘𝑂)𝑦)
19	18	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑢 ∧ 𝑥 ∈ 𝑡) → (𝑦(+g‘𝐺)𝑥) = (𝑥(+g‘𝑂)𝑦))
20	19	mpoeq3ia 7090	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑥(+g‘𝑂)𝑦))
21	20	tposmpo 7780	. . . . . . . . . . 11 ⊢ tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))
22	21	rneqi 5689	. . . . . . . . . 10 ⊢ ran tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))
23	14, 22	eqtr3i 2821	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))
24	23	a1i 11	. . . . . . . 8 ⊢ ((𝑢 ∈ 𝒫 (Base‘𝐺) ∧ 𝑡 ∈ 𝒫 (Base‘𝐺)) → ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)))
25	24	mpoeq3ia 7090	. . . . . . 7 ⊢ (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))) = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)))
26	25	tposmpo 7780	. . . . . 6 ⊢ tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)))
27	5, 26	syl6eq 2847	. . . . 5 ⊢ (𝐺 ∈ V → tpos ⊕ = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))))
28	15	fvexi 6552	. . . . . 6 ⊢ 𝑂 ∈ V
29	15, 1	oppgbas 18220	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝑂)
30		eqid 2795	. . . . . . 7 ⊢ (LSSum‘𝑂) = (LSSum‘𝑂)
31	29, 16, 30	lsmfval 18493	. . . . . 6 ⊢ (𝑂 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))))
32	28, 31	ax-mp 5	. . . . 5 ⊢ (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)))
33	27, 32	syl6reqr 2850	. . . 4 ⊢ (𝐺 ∈ V → (LSSum‘𝑂) = tpos ⊕ )
34	33	oveqd 7033	. . 3 ⊢ (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇tpos ⊕ 𝑈))
35		ovtpos 7758	. . 3 ⊢ (𝑇tpos ⊕ 𝑈) = (𝑈 ⊕ 𝑇)
36	34, 35	syl6eq 2847	. 2 ⊢ (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇))
37		eqid 2795	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)
38		0ex 5102	. . . . . . 7 ⊢ ∅ ∈ V
39		eqidd 2796	. . . . . . 7 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → ∅ = ∅)
40	37, 38, 39	elovmpo 7249	. . . . . 6 ⊢ (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ↔ (𝑇 ∈ 𝒫 (Base‘𝐺) ∧ 𝑈 ∈ 𝒫 (Base‘𝐺) ∧ 𝑥 ∈ ∅))
41	40	simp3bi 1140	. . . . 5 ⊢ (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) → 𝑥 ∈ ∅)
42	41	ssriv 3893	. . . 4 ⊢ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅
43		ss0 4272	. . . 4 ⊢ ((𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅ → (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) = ∅)
44	42, 43	ax-mp 5	. . 3 ⊢ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) = ∅
45		elpwi 4463	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝒫 (Base‘𝐺) → 𝑡 ⊆ (Base‘𝐺))
46	45	3ad2ant2 1127	. . . . . . . . . . . 12 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ (Base‘𝐺))
47		fvprc 6531	. . . . . . . . . . . . 13 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
48	47	3ad2ant1 1126	. . . . . . . . . . . 12 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (Base‘𝐺) = ∅)
49	46, 48	sseqtrd 3928	. . . . . . . . . . 11 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ ∅)
50		ss0 4272	. . . . . . . . . . 11 ⊢ (𝑡 ⊆ ∅ → 𝑡 = ∅)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 = ∅)
52		eqid 2795	. . . . . . . . . 10 ⊢ 𝑢 = 𝑢
53		mpoeq12 7085	. . . . . . . . . 10 ⊢ ((𝑡 = ∅ ∧ 𝑢 = 𝑢) → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)))
54	51, 52, 53	sylancl 586	. . . . . . . . 9 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = (𝑥 ∈ ∅, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)))
55		mpo0 7095	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ∅
56	54, 55	syl6eq 2847	. . . . . . . 8 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ∅)
57	56	rneqd 5690	. . . . . . 7 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ran ∅)
58		rn0 5715	. . . . . . 7 ⊢ ran ∅ = ∅
59	57, 58	syl6eq 2847	. . . . . 6 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ∅)
60	59	mpoeq3dva 7089	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
61	32, 60	syl5eq 2843	. . . 4 ⊢ (¬ 𝐺 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
62	61	oveqd 7033	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈))
63		fvprc 6531	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (LSSum‘𝐺) = ∅)
64	3, 63	syl5eq 2843	. . . . 5 ⊢ (¬ 𝐺 ∈ V → ⊕ = ∅)
65	64	oveqd 7033	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑈 ⊕ 𝑇) = (𝑈∅𝑇))
66		0ov 7052	. . . 4 ⊢ (𝑈∅𝑇) = ∅
67	65, 66	syl6eq 2847	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑈 ⊕ 𝑇) = ∅)
68	44, 62, 67	3eqtr4a 2857	. 2 ⊢ (¬ 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇))
69	36, 68	pm2.61i 183	1 ⊢ (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇)

Syntax hints:  ¬ wn 3   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ⊆ wss 3859  ∅c0 4211  𝒫 cpw 4453  ^◡ccnv 5442  dom cdm 5443  ran crn 5444  Rel wrel 5448  Fun wfun 6219  –onto→wfo 6223  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018  tpos ctpos 7742  Basecbs 16312  +_gcplusg 16394  opp_gcoppg 18214  LSSumclsm 18489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-tpos 7743  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-plusg 16407  df-oppg 18215  df-lsm 18491

This theorem is referenced by:  lsmmod2  18529  lsmdisj2r  18538