exsslsb - Mathbox for Thierry Arnoux

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Theorem exsslsb 33793

Description: Any finite generating set 𝑆 of a vector space 𝑊 contains a basis. (Contributed by Thierry Arnoux, 13-Oct-2025.)

**Hypotheses**
Ref	Expression
exsslsb.b	⊢ 𝐵 = (Base‘𝑊)
exsslsb.j	⊢ 𝐽 = (LBasis‘𝑊)
exsslsb.k	⊢ 𝐾 = (LSpan‘𝑊)
exsslsb.w	⊢ (𝜑 → 𝑊 ∈ LVec)
exsslsb.s	⊢ (𝜑 → 𝑆 ∈ Fin)
exsslsb.1	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
exsslsb.2	⊢ (𝜑 → (𝐾‘𝑆) = 𝐵)

**Assertion**
Ref	Expression
exsslsb	⊢ (𝜑 → ∃𝑠 ∈ 𝐽 𝑠 ⊆ 𝑆)

Distinct variable groups: 𝐵,𝑠 𝐽,𝑠 𝐾,𝑠 𝑆,𝑠 𝜑,𝑠 Allowed substitution hint: 𝑊(𝑠)

Proof of Theorem exsslsb Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1922	. 2 ⊢ Ⅎ𝑠𝜑
2		exsslsb.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec)
3	2	ad2antrr 733	. . 3 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑊 ∈ LVec)
4		simplr 775	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))))
5	4	elin2d 4137	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))
6	5	elin1d 4136	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ∈ 𝒫 𝑆)
7	6	elpwid 4541	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ⊆ 𝑆)
8		exsslsb.1	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
9	8	ad2antrr 733	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑆 ⊆ 𝐵)
10	7, 9	sstrd 3927	. . 3 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ⊆ 𝐵)
11		lveclmod 21100	. . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
12		exsslsb.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊)
13		eqid 2741	. . . . . . . 8 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
14		exsslsb.k	. . . . . . . 8 ⊢ 𝐾 = (LSpan‘𝑊)
15	12, 13, 14	lspf 20968	. . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐾:𝒫 𝐵⟶(LSubSp‘𝑊))
16	2, 11, 15	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶(LSubSp‘𝑊))
17	16	ffnd 6660	. . . . 5 ⊢ (𝜑 → 𝐾 Fn 𝒫 𝐵)
18	17	ad2antrr 733	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝐾 Fn 𝒫 𝐵)
19	5	elin2d 4137	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ∈ (^◡𝐾 “ {𝐵}))
20		fniniseg 7005	. . . . 5 ⊢ (𝐾 Fn 𝒫 𝐵 → (𝑠 ∈ (^◡𝐾 “ {𝐵}) ↔ (𝑠 ∈ 𝒫 𝐵 ∧ (𝐾‘𝑠) = 𝐵)))
21	20	simplbda 501	. . . 4 ⊢ ((𝐾 Fn 𝒫 𝐵 ∧ 𝑠 ∈ (^◡𝐾 “ {𝐵})) → (𝐾‘𝑠) = 𝐵)
22	18, 19, 21	syl2anc 591	. . 3 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → (𝐾‘𝑠) = 𝐵)
23	2, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ LMod)
24	23	ad3antrrr 737	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑊 ∈ LMod)
25		simpr 486	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ⊊ 𝑠)
26	25	pssssd 4034	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ⊆ 𝑠)
27	7	adantr 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑠 ⊆ 𝑆)
28	26, 27	sstrd 3927	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ⊆ 𝑆)
29	9	adantr 482	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑆 ⊆ 𝐵)
30	28, 29	sstrd 3927	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ⊆ 𝐵)
31	12, 14	lspssv 20977	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑢 ⊆ 𝐵) → (𝐾‘𝑢) ⊆ 𝐵)
32	24, 30, 31	syl2anc 591	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → (𝐾‘𝑢) ⊆ 𝐵)
33		hashf 14295	. . . . . . . . . . . 12 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
34		ffun 6662	. . . . . . . . . . . 12 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → Fun ♯)
35	33, 34	mp1i 13	. . . . . . . . . . 11 ⊢ (𝜑 → Fun ♯)
36		exsslsb.s	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 ∈ Fin)
37		pwssfi 9105	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∈ Fin → (𝑆 ∈ Fin ↔ 𝒫 𝑆 ⊆ Fin))
38	37	ibi 269	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ Fin → 𝒫 𝑆 ⊆ Fin)
39	36, 38	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝒫 𝑆 ⊆ Fin)
40	39	ssinss1d 4178	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})) ⊆ Fin)
41	40	sselda 3917	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) → 𝑠 ∈ Fin)
42		hashcl 14313	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → (♯‘𝑠) ∈ ℕ₀)
43	41, 42	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) → (♯‘𝑠) ∈ ℕ₀)
44		nn0uz 12821	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
45	43, 44	eleqtrdi 2851	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) → (♯‘𝑠) ∈ (ℤ_≥‘0))
46	1, 35, 45	funimassd 6897	. . . . . . . . . 10 ⊢ (𝜑 → (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) ⊆ (ℤ_≥‘0))
47	46	ad3antrrr 737	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) ⊆ (ℤ_≥‘0))
48	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ♯:V⟶(ℕ₀ ∪ {+∞}))
49	48	ffnd 6660	. . . . . . . . . . . 12 ⊢ (𝜑 → ♯ Fn V)
50	49	ad3antrrr 737	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → ♯ Fn V)
51	50	adantr 482	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → ♯ Fn V)
52		vex 3437	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
53	52	a1i 11	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑢 ∈ V)
54	36	ad3antrrr 737	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑆 ∈ Fin)
55	54, 28	sselpwd 5259	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ∈ 𝒫 𝑆)
56	55	adantr 482	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑢 ∈ 𝒫 𝑆)
57	18	ad2antrr 733	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝐾 Fn 𝒫 𝐵)
58	12	fvexi 6845	. . . . . . . . . . . . . . 15 ⊢ 𝐵 ∈ V
59	58	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝐵 ∈ V)
60	59, 30	sselpwd 5259	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ∈ 𝒫 𝐵)
61	60	adantr 482	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑢 ∈ 𝒫 𝐵)
62		simpr 486	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (𝐾‘𝑢) = 𝐵)
63		fvex 6844	. . . . . . . . . . . . . 14 ⊢ (𝐾‘𝑢) ∈ V
64	63	elsn 4573	. . . . . . . . . . . . 13 ⊢ ((𝐾‘𝑢) ∈ {𝐵} ↔ (𝐾‘𝑢) = 𝐵)
65	62, 64	sylibr 236	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (𝐾‘𝑢) ∈ {𝐵})
66	57, 61, 65	elpreimad 7004	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑢 ∈ (^◡𝐾 “ {𝐵}))
67	56, 66	elind 4132	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑢 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))
68	51, 53, 67	fnfvimad 7182	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (♯‘𝑢) ∈ (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))))
69		infssuzle 12876	. . . . . . . . 9 ⊢ (((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) ⊆ (ℤ_≥‘0) ∧ (♯‘𝑢) ∈ (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) → inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ≤ (♯‘𝑢))
70	47, 68, 69	syl2an2r 692	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ≤ (♯‘𝑢))
71	54, 27	ssfid 9173	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑠 ∈ Fin)
72	71	adantr 482	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑠 ∈ Fin)
73		simplr 775	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑢 ⊊ 𝑠)
74		hashpss 32905	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ Fin ∧ 𝑢 ⊊ 𝑠) → (♯‘𝑢) < (♯‘𝑠))
75	72, 73, 74	syl2anc 591	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (♯‘𝑢) < (♯‘𝑠))
76		simpllr 782	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ))
77	75, 76	breqtrd 5101	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (♯‘𝑢) < inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ))
78	26	adantr 482	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑢 ⊆ 𝑠)
79	72, 78	ssfid 9173	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑢 ∈ Fin)
80		hashcl 14313	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ Fin → (♯‘𝑢) ∈ ℕ₀)
81	79, 80	syl 17	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (♯‘𝑢) ∈ ℕ₀)
82	81	nn0red 12494	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (♯‘𝑢) ∈ ℝ)
83	72, 42	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (♯‘𝑠) ∈ ℕ₀)
84	83	nn0red 12494	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (♯‘𝑠) ∈ ℝ)
85	76, 84	eqeltrrd 2842	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ∈ ℝ)
86	82, 85	ltnled 11288	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → ((♯‘𝑢) < inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ↔ ¬ inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ≤ (♯‘𝑢)))
87	77, 86	mpbid 234	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → ¬ inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ≤ (♯‘𝑢))
88	70, 87	pm2.65da 823	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → ¬ (𝐾‘𝑢) = 𝐵)
89	88	neqned 2943	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → (𝐾‘𝑢) ≠ 𝐵)
90		df-pss 3905	. . . . . 6 ⊢ ((𝐾‘𝑢) ⊊ 𝐵 ↔ ((𝐾‘𝑢) ⊆ 𝐵 ∧ (𝐾‘𝑢) ≠ 𝐵))
91	32, 89, 90	sylanbrc 590	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → (𝐾‘𝑢) ⊊ 𝐵)
92	91	ex 414	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → (𝑢 ⊊ 𝑠 → (𝐾‘𝑢) ⊊ 𝐵))
93	92	alrimiv 1935	. . 3 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → ∀𝑢(𝑢 ⊊ 𝑠 → (𝐾‘𝑢) ⊊ 𝐵))
94		exsslsb.j	. . . . 5 ⊢ 𝐽 = (LBasis‘𝑊)
95	12, 94, 14	islbs3 21152	. . . 4 ⊢ (𝑊 ∈ LVec → (𝑠 ∈ 𝐽 ↔ (𝑠 ⊆ 𝐵 ∧ (𝐾‘𝑠) = 𝐵 ∧ ∀𝑢(𝑢 ⊊ 𝑠 → (𝐾‘𝑢) ⊊ 𝐵))))
96	95	biimpar 479	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝑠 ⊆ 𝐵 ∧ (𝐾‘𝑠) = 𝐵 ∧ ∀𝑢(𝑢 ⊊ 𝑠 → (𝐾‘𝑢) ⊊ 𝐵))) → 𝑠 ∈ 𝐽)
97	3, 10, 22, 93, 96	syl13anc 1381	. 2 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ∈ 𝐽)
98	36	elexd 3456	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
99		pwidg 4552	. . . . . . . 8 ⊢ (𝑆 ∈ Fin → 𝑆 ∈ 𝒫 𝑆)
100	36, 99	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝑆)
101	36, 8	elpwd 4538	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
102		exsslsb.2	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝑆) = 𝐵)
103		fvex 6844	. . . . . . . . . 10 ⊢ (𝐾‘𝑆) ∈ V
104	103	elsn 4573	. . . . . . . . 9 ⊢ ((𝐾‘𝑆) ∈ {𝐵} ↔ (𝐾‘𝑆) = 𝐵)
105	102, 104	sylibr 236	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘𝑆) ∈ {𝐵})
106	17, 101, 105	elpreimad 7004	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (^◡𝐾 “ {𝐵}))
107	100, 106	elind 4132	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))
108	49, 98, 107	fnfvimad 7182	. . . . 5 ⊢ (𝜑 → (♯‘𝑆) ∈ (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))))
109	108	ne0d 4273	. . . 4 ⊢ (𝜑 → (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) ≠ ∅)
110		infssuzcl 12877	. . . 4 ⊢ (((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) ⊆ (ℤ_≥‘0) ∧ (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) ≠ ∅) → inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ∈ (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))))
111	46, 109, 110	syl2anc 591	. . 3 ⊢ (𝜑 → inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ∈ (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))))
112		fvelima2 6883	. . 3 ⊢ ((♯ Fn V ∧ inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ∈ (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) → ∃𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))(♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ))
113	49, 111, 112	syl2anc 591	. 2 ⊢ (𝜑 → ∃𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))(♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ))
114	1, 97, 7, 113	reximd2a 3251	1 ⊢ (𝜑 → ∃𝑠 ∈ 𝐽 𝑠 ⊆ 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∧ w3a 1093 ∀wal 1546 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∃wrex 3065 Vcvv 3433 ∪ cun 3883 ∩ cin 3884 ⊆ wss 3885 ⊊ wpss 3886 ∅c0 4264 𝒫 cpw 4532 {csn 4558 class class class wbr 5075 ^◡ccnv 5620 “ cima 5624 Fun wfun 6483 Fn wfn 6484 ⟶wf 6485 ‘cfv 6489 Fincfn 8887 infcinf 9348 ℝcr 11032 0cc0 11033 +∞cpnf 11171 < clt 11174 ≤ cle 11175 ℕ₀cn0 12432 ℤ_≥cuz 12783 ♯chash 14287 Basecbs 17174 LModclmod 20854 LSubSpclss 20925 LSpanclspn 20965 LBasisclbs 21068 LVecclvec 21096

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 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-xnn0 12506 df-z 12520 df-uz 12784 df-fz 13457 df-hash 14288 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-0g 17399 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18907 df-minusg 18908 df-sbg 18909 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-drng 20707 df-lmod 20856 df-lss 20926 df-lsp 20966 df-lbs 21069 df-lvec 21097

This theorem is referenced by: lbslelsp 33794

W3C validator