Theorem oppglsm 19683

Description: The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.) (Proof shortened by AV, 2-Mar-2024.)

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		oppglsm.o	. . . . . . 7 ⊢ 𝑂 = (oppg‘𝐺)
2	1	fvexi 6882	. . . . . 6 ⊢ 𝑂 ∈ V
3		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	1, 3	oppgbas 19392	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝑂)
5		eqid 2763	. . . . . . 7 ⊢ (+g‘𝑂) = (+g‘𝑂)
6		eqid 2763	. . . . . . 7 ⊢ (LSSum‘𝑂) = (LSSum‘𝑂)
7	4, 5, 6	lsmfval 19679	. . . . . 6 ⊢ (𝑂 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))))
8	2, 7	ax-mp 5	. . . . 5 ⊢ (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)))
9		eqid 2763	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10		oppglsm.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝐺)
11	3, 9, 10	lsmfval 19679	. . . . . . 7 ⊢ (𝐺 ∈ V → ⊕ = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))))
12	11	tposeqd 8210	. . . . . 6 ⊢ (𝐺 ∈ V → tpos ⊕ = tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))))
13		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
14	13	reldmmpo 7531	. . . . . . . . . . . 12 ⊢ Rel dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
15	13	mpofun 7521	. . . . . . . . . . . . 13 ⊢ Fun (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
16		funforn 6786	. . . . . . . . . . . . 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)))
17	15, 16	mpbi 232	. . . . . . . . . . . 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))
18		tposfo2 8230	. . . . . . . . . . . 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))))
19	14, 17, 18	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))
20		forn 6782	. . . . . . . . . . 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)))
21	19, 20	ax-mp 5	. . . . . . . . . 10 ⊢ ran tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
22	9, 1, 5	oppgplus 19390	. . . . . . . . . . . . . . 15 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑥)
23	22	eqcomi 2772	. . . . . . . . . . . . . 14 ⊢ (𝑦(+g‘𝐺)𝑥) = (𝑥(+g‘𝑂)𝑦)
24	23	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑢 ∧ 𝑥 ∈ 𝑡) → (𝑦(+g‘𝐺)𝑥) = (𝑥(+g‘𝑂)𝑦))
25	24	mpoeq3ia 7475	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑥(+g‘𝑂)𝑦))
26	25	tposmpo 8244	. . . . . . . . . . 11 ⊢ tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))
27	26	rneqi 5914	. . . . . . . . . 10 ⊢ ran tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))
28	21, 27	eqtr3i 2788	. . . . . . . . 9 ⊢ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))
29	28	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)))
30	29	mpoeq3ia 7475	. . . . . . 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)))
31	30	tposmpo 8244	. . . . . 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)))
32	12, 31	eqtrdi 2814	. . . . 5 ⊢ (𝐺 ∈ V → tpos ⊕ = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))))
33	8, 32	eqtr4id 2817	. . . 4 ⊢ (𝐺 ∈ V → (LSSum‘𝑂) = tpos ⊕ )
34	33	oveqd 7414	. . 3 ⊢ (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇tpos ⊕ 𝑈))
35		ovtpos 8222	. . 3 ⊢ (𝑇tpos ⊕ 𝑈) = (𝑈 ⊕ 𝑇)
36	34, 35	eqtrdi 2814	. 2 ⊢ (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇))
37		eqid 2763	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)
38		0ex 5258	. . . . . . 7 ⊢ ∅ ∈ V
39		eqidd 2764	. . . . . . 7 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → ∅ = ∅)
40	37, 38, 39	elovmpo 7642	. . . . . 6 ⊢ (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ↔ (𝑇 ∈ 𝒫 (Base‘𝐺) ∧ 𝑈 ∈ 𝒫 (Base‘𝐺) ∧ 𝑥 ∈ ∅))
41	40	simp3bi 1161	. . . . 5 ⊢ (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) → 𝑥 ∈ ∅)
42	41	ssriv 3941	. . . 4 ⊢ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅
43		ss0 4357	. . . 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 4563	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝒫 (Base‘𝐺) → 𝑡 ⊆ (Base‘𝐺))
46	45	3ad2ant2 1148	. . . . . . . . . . . 12 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ (Base‘𝐺))
47		fvprc 6860	. . . . . . . . . . . . 13 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
48	47	3ad2ant1 1147	. . . . . . . . . . . 12 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (Base‘𝐺) = ∅)
49	46, 48	sseqtrd 3973	. . . . . . . . . . 11 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ ∅)
50		ss0 4357	. . . . . . . . . . 11 ⊢ (𝑡 ⊆ ∅ → 𝑡 = ∅)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 = ∅)
52	51	orcd 884	. . . . . . . . 9 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑡 = ∅ ∨ 𝑢 = ∅))
53		0mpo0 7480	. . . . . . . . 9 ⊢ ((𝑡 = ∅ ∨ 𝑢 = ∅) → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ∅)
54	52, 53	syl 17	. . . . . . . 8 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ∅)
55	54	rneqd 5915	. . . . . . 7 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ran ∅)
56		rn0 5903	. . . . . . 7 ⊢ ran ∅ = ∅
57	55, 56	eqtrdi 2814	. . . . . 6 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ∅)
58	57	mpoeq3dva 7474	. . . . 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) ↦ ∅))
59	8, 58	eqtrid 2810	. . . 4 ⊢ (¬ 𝐺 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
60	59	oveqd 7414	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈))
61		fvprc 6860	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (LSSum‘𝐺) = ∅)
62	10, 61	eqtrid 2810	. . . . 5 ⊢ (¬ 𝐺 ∈ V → ⊕ = ∅)
63	62	oveqd 7414	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑈 ⊕ 𝑇) = (𝑈∅𝑇))
64		0ov 7434	. . . 4 ⊢ (𝑈∅𝑇) = ∅
65	63, 64	eqtrdi 2814	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑈 ⊕ 𝑇) = ∅)
66	44, 60, 65	3eqtr4a 2824	. 2 ⊢ (¬ 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇))
67	36, 66	pm2.61i 183	1 ⊢ (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇)

Syntax hints:  ¬ wn 3   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4556  ^◡ccnv 5647  dom cdm 5648  ran crn 5649  Rel wrel 5653  Fun wfun 6516  –onto→wfo 6520  ‘cfv 6522  (class class class)co 7397   ∈ cmpo 7399  tpos ctpos 8206  Basecbs 17246  +_gcplusg 17287  opp_gcoppg 19386  LSSumclsm 19675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-1st 7971  df-2nd 7972  df-tpos 8207  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-er 8679  df-en 8929  df-dom 8930  df-sdom 8931  df-pnf 11219  df-mnf 11220  df-xr 11221  df-ltxr 11222  df-le 11223  df-sub 11417  df-neg 11418  df-nn 12212  df-2 12281  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17247  df-plusg 17300  df-oppg 19387  df-lsm 19677

This theorem is referenced by:  lsmmod2  19717  lsmdisj2r  19726  lsmsnorb2  33579