Theorem oppglsm 19424

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 6856	. . . . . 6 ⊢ 𝑂 ∈ V
3		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	1, 3	oppgbas 19130	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝑂)
5		eqid 2736	. . . . . . 7 ⊢ (+g‘𝑂) = (+g‘𝑂)
6		eqid 2736	. . . . . . 7 ⊢ (LSSum‘𝑂) = (LSSum‘𝑂)
7	4, 5, 6	lsmfval 19420	. . . . . 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 2736	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10		oppglsm.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝐺)
11	3, 9, 10	lsmfval 19420	. . . . . . 7 ⊢ (𝐺 ∈ V → ⊕ = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))))
12	11	tposeqd 8160	. . . . . 6 ⊢ (𝐺 ∈ V → tpos ⊕ = tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))))
13		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
14	13	reldmmpo 7490	. . . . . . . . . . . 12 ⊢ Rel dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
15	13	mpofun 7480	. . . . . . . . . . . . 13 ⊢ Fun (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
16		funforn 6763	. . . . . . . . . . . . 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 229	. . . . . . . . . . . 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 8180	. . . . . . . . . . . 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 6759	. . . . . . . . . . 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 19127	. . . . . . . . . . . . . . 15 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑥)
23	22	eqcomi 2745	. . . . . . . . . . . . . 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 7435	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑥(+g‘𝑂)𝑦))
26	25	tposmpo 8194	. . . . . . . . . . 11 ⊢ tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))
27	26	rneqi 5892	. . . . . . . . . 10 ⊢ ran tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))
28	21, 27	eqtr3i 2766	. . . . . . . . 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 7435	. . . . . . 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 8194	. . . . . 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 2792	. . . . 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 2795	. . . 4 ⊢ (𝐺 ∈ V → (LSSum‘𝑂) = tpos ⊕ )
34	33	oveqd 7374	. . 3 ⊢ (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇tpos ⊕ 𝑈))
35		ovtpos 8172	. . 3 ⊢ (𝑇tpos ⊕ 𝑈) = (𝑈 ⊕ 𝑇)
36	34, 35	eqtrdi 2792	. 2 ⊢ (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇))
37		eqid 2736	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)
38		0ex 5264	. . . . . . 7 ⊢ ∅ ∈ V
39		eqidd 2737	. . . . . . 7 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → ∅ = ∅)
40	37, 38, 39	elovmpo 7598	. . . . . 6 ⊢ (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ↔ (𝑇 ∈ 𝒫 (Base‘𝐺) ∧ 𝑈 ∈ 𝒫 (Base‘𝐺) ∧ 𝑥 ∈ ∅))
41	40	simp3bi 1147	. . . . 5 ⊢ (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) → 𝑥 ∈ ∅)
42	41	ssriv 3948	. . . 4 ⊢ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅
43		ss0 4358	. . . 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 4567	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝒫 (Base‘𝐺) → 𝑡 ⊆ (Base‘𝐺))
46	45	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ (Base‘𝐺))
47		fvprc 6834	. . . . . . . . . . . . 13 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
48	47	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (Base‘𝐺) = ∅)
49	46, 48	sseqtrd 3984	. . . . . . . . . . 11 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ ∅)
50		ss0 4358	. . . . . . . . . . 11 ⊢ (𝑡 ⊆ ∅ → 𝑡 = ∅)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 = ∅)
52	51	orcd 871	. . . . . . . . 9 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑡 = ∅ ∨ 𝑢 = ∅))
53		0mpo0 7440	. . . . . . . . 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 5893	. . . . . . 7 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ran ∅)
56		rn0 5881	. . . . . . 7 ⊢ ran ∅ = ∅
57	55, 56	eqtrdi 2792	. . . . . 6 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ∅)
58	57	mpoeq3dva 7434	. . . . 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 2788	. . . 4 ⊢ (¬ 𝐺 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
60	59	oveqd 7374	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈))
61		fvprc 6834	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (LSSum‘𝐺) = ∅)
62	10, 61	eqtrid 2788	. . . . 5 ⊢ (¬ 𝐺 ∈ V → ⊕ = ∅)
63	62	oveqd 7374	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑈 ⊕ 𝑇) = (𝑈∅𝑇))
64		0ov 7394	. . . 4 ⊢ (𝑈∅𝑇) = ∅
65	63, 64	eqtrdi 2792	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑈 ⊕ 𝑇) = ∅)
66	44, 60, 65	3eqtr4a 2802	. 2 ⊢ (¬ 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇))
67	36, 66	pm2.61i 182	1 ⊢ (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇)

Syntax hints:  ¬ wn 3   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  ^◡ccnv 5632  dom cdm 5633  ran crn 5634  Rel wrel 5638  Fun wfun 6490  –onto→wfo 6494  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  tpos ctpos 8156  Basecbs 17083  +_gcplusg 17133  opp_gcoppg 19123  LSSumclsm 19416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-oppg 19124  df-lsm 19418

This theorem is referenced by:  lsmmod2  19458  lsmdisj2r  19467  lsmsnorb2  32173