Theorem oppglsm 19611

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 6849	. . . . . 6 ⊢ 𝑂 ∈ V
3		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	1, 3	oppgbas 19320	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝑂)
5		eqid 2737	. . . . . . 7 ⊢ (+g‘𝑂) = (+g‘𝑂)
6		eqid 2737	. . . . . . 7 ⊢ (LSSum‘𝑂) = (LSSum‘𝑂)
7	4, 5, 6	lsmfval 19607	. . . . . 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 2737	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
10		oppglsm.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝐺)
11	3, 9, 10	lsmfval 19607	. . . . . . 7 ⊢ (𝐺 ∈ V → ⊕ = (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))))
12	11	tposeqd 8173	. . . . . 6 ⊢ (𝐺 ∈ V → tpos ⊕ = tpos (𝑢 ∈ 𝒫 (Base‘𝐺), 𝑡 ∈ 𝒫 (Base‘𝐺) ↦ ran (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))))
13		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
14	13	reldmmpo 7495	. . . . . . . . . . . 12 ⊢ Rel dom (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
15	13	mpofun 7485	. . . . . . . . . . . . 13 ⊢ Fun (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥))
16		funforn 6754	. . . . . . . . . . . . 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 230	. . . . . . . . . . . 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 8193	. . . . . . . . . . . 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 6750	. . . . . . . . . . 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 19318	. . . . . . . . . . . . . . 15 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑥)
23	22	eqcomi 2746	. . . . . . . . . . . . . 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 7439	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑥(+g‘𝑂)𝑦))
26	25	tposmpo 8207	. . . . . . . . . . 11 ⊢ tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))
27	26	rneqi 5887	. . . . . . . . . 10 ⊢ ran tpos (𝑦 ∈ 𝑢, 𝑥 ∈ 𝑡 ↦ (𝑦(+g‘𝐺)𝑥)) = ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦))
28	21, 27	eqtr3i 2762	. . . . . . . . 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 7439	. . . . . . 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 8207	. . . . . 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 2788	. . . . 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 2791	. . . 4 ⊢ (𝐺 ∈ V → (LSSum‘𝑂) = tpos ⊕ )
34	33	oveqd 7378	. . 3 ⊢ (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇tpos ⊕ 𝑈))
35		ovtpos 8185	. . 3 ⊢ (𝑇tpos ⊕ 𝑈) = (𝑈 ⊕ 𝑇)
36	34, 35	eqtrdi 2788	. 2 ⊢ (𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑈 ⊕ 𝑇))
37		eqid 2737	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)
38		0ex 5243	. . . . . . 7 ⊢ ∅ ∈ V
39		eqidd 2738	. . . . . . 7 ⊢ ((𝑡 = 𝑇 ∧ 𝑢 = 𝑈) → ∅ = ∅)
40	37, 38, 39	elovmpo 7606	. . . . . 6 ⊢ (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ↔ (𝑇 ∈ 𝒫 (Base‘𝐺) ∧ 𝑈 ∈ 𝒫 (Base‘𝐺) ∧ 𝑥 ∈ ∅))
41	40	simp3bi 1148	. . . . 5 ⊢ (𝑥 ∈ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) → 𝑥 ∈ ∅)
42	41	ssriv 3926	. . . 4 ⊢ (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈) ⊆ ∅
43		ss0 4343	. . . 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 4549	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝒫 (Base‘𝐺) → 𝑡 ⊆ (Base‘𝐺))
46	45	3ad2ant2 1135	. . . . . . . . . . . 12 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ (Base‘𝐺))
47		fvprc 6827	. . . . . . . . . . . . 13 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
48	47	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (Base‘𝐺) = ∅)
49	46, 48	sseqtrd 3959	. . . . . . . . . . 11 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 ⊆ ∅)
50		ss0 4343	. . . . . . . . . . 11 ⊢ (𝑡 ⊆ ∅ → 𝑡 = ∅)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → 𝑡 = ∅)
52	51	orcd 874	. . . . . . . . 9 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → (𝑡 = ∅ ∨ 𝑢 = ∅))
53		0mpo0 7444	. . . . . . . . 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 5888	. . . . . . 7 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ran ∅)
56		rn0 5876	. . . . . . 7 ⊢ ran ∅ = ∅
57	55, 56	eqtrdi 2788	. . . . . 6 ⊢ ((¬ 𝐺 ∈ V ∧ 𝑡 ∈ 𝒫 (Base‘𝐺) ∧ 𝑢 ∈ 𝒫 (Base‘𝐺)) → ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑂)𝑦)) = ∅)
58	57	mpoeq3dva 7438	. . . . 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 2784	. . . 4 ⊢ (¬ 𝐺 ∈ V → (LSSum‘𝑂) = (𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅))
60	59	oveqd 7378	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑇(LSSum‘𝑂)𝑈) = (𝑇(𝑡 ∈ 𝒫 (Base‘𝐺), 𝑢 ∈ 𝒫 (Base‘𝐺) ↦ ∅)𝑈))
61		fvprc 6827	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (LSSum‘𝐺) = ∅)
62	10, 61	eqtrid 2784	. . . . 5 ⊢ (¬ 𝐺 ∈ V → ⊕ = ∅)
63	62	oveqd 7378	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑈 ⊕ 𝑇) = (𝑈∅𝑇))
64		0ov 7398	. . . 4 ⊢ (𝑈∅𝑇) = ∅
65	63, 64	eqtrdi 2788	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑈 ⊕ 𝑇) = ∅)
66	44, 60, 65	3eqtr4a 2798	. 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 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ^◡ccnv 5624  dom cdm 5625  ran crn 5626  Rel wrel 5630  Fun wfun 6487  –onto→wfo 6491  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363  tpos ctpos 8169  Basecbs 17173  +_gcplusg 17214  opp_gcoppg 19314  LSSumclsm 19603

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-tpos 8170  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-plusg 17227  df-oppg 19315  df-lsm 19605

This theorem is referenced by:  lsmmod2  19645  lsmdisj2r  19654  lsmsnorb2  33470