exsslsb - Mathbox for Thierry Arnoux

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Theorem exsslsb 33781

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 1921	. 2 ⊢ Ⅎ𝑠𝜑
2		exsslsb.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec)
3	2	ad2antrr 732	. . 3 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑊 ∈ LVec)
4		simplr 774	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))))
5	4	elin2d 4134	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))
6	5	elin1d 4133	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ∈ 𝒫 𝑆)
7	6	elpwid 4538	. . . 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 732	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑆 ⊆ 𝐵)
10	7, 9	sstrd 3925	. . 3 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ⊆ 𝐵)
11		lveclmod 21096	. . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
12		exsslsb.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊)
13		eqid 2739	. . . . . . . 8 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊)
14		exsslsb.k	. . . . . . . 8 ⊢ 𝐾 = (LSpan‘𝑊)
15	12, 13, 14	lspf 20964	. . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝐾:𝒫 𝐵⟶(LSubSp‘𝑊))
16	2, 11, 15	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶(LSubSp‘𝑊))
17	16	ffnd 6656	. . . . 5 ⊢ (𝜑 → 𝐾 Fn 𝒫 𝐵)
18	17	ad2antrr 732	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝐾 Fn 𝒫 𝐵)
19	5	elin2d 4134	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ∈ (^◡𝐾 “ {𝐵}))
20		fniniseg 7001	. . . . 5 ⊢ (𝐾 Fn 𝒫 𝐵 → (𝑠 ∈ (^◡𝐾 “ {𝐵}) ↔ (𝑠 ∈ 𝒫 𝐵 ∧ (𝐾‘𝑠) = 𝐵)))
21	20	simplbda 500	. . . 4 ⊢ ((𝐾 Fn 𝒫 𝐵 ∧ 𝑠 ∈ (^◡𝐾 “ {𝐵})) → (𝐾‘𝑠) = 𝐵)
22	18, 19, 21	syl2anc 590	. . 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 736	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑊 ∈ LMod)
25		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ⊊ 𝑠)
26	25	pssssd 4031	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ⊆ 𝑠)
27	7	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑠 ⊆ 𝑆)
28	26, 27	sstrd 3925	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ⊆ 𝑆)
29	9	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑆 ⊆ 𝐵)
30	28, 29	sstrd 3925	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ⊆ 𝐵)
31	12, 14	lspssv 20973	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑢 ⊆ 𝐵) → (𝐾‘𝑢) ⊆ 𝐵)
32	24, 30, 31	syl2anc 590	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → (𝐾‘𝑢) ⊆ 𝐵)
33		hashf 14291	. . . . . . . . . . . 12 ⊢ ♯:V⟶(ℕ₀ ∪ {+∞})
34		ffun 6658	. . . . . . . . . . . 12 ⊢ (♯:V⟶(ℕ₀ ∪ {+∞}) → Fun ♯)
35	33, 34	mp1i 13	. . . . . . . . . . 11 ⊢ (𝜑 → Fun ♯)
36		exsslsb.s	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 ∈ Fin)
37		pwssfi 9101	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∈ Fin → (𝑆 ∈ Fin ↔ 𝒫 𝑆 ⊆ Fin))
38	37	ibi 268	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ Fin → 𝒫 𝑆 ⊆ Fin)
39	36, 38	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝒫 𝑆 ⊆ Fin)
40	39	ssinss1d 4175	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})) ⊆ Fin)
41	40	sselda 3915	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) → 𝑠 ∈ Fin)
42		hashcl 14309	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → (♯‘𝑠) ∈ ℕ₀)
43	41, 42	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) → (♯‘𝑠) ∈ ℕ₀)
44		nn0uz 12817	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
45	43, 44	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) → (♯‘𝑠) ∈ (ℤ_≥‘0))
46	1, 35, 45	funimassd 6893	. . . . . . . . . 10 ⊢ (𝜑 → (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) ⊆ (ℤ_≥‘0))
47	46	ad3antrrr 736	. . . . . . . . 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 6656	. . . . . . . . . . . 12 ⊢ (𝜑 → ♯ Fn V)
50	49	ad3antrrr 736	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → ♯ Fn V)
51	50	adantr 481	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → ♯ Fn V)
52		vex 3435	. . . . . . . . . . 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 736	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑆 ∈ Fin)
55	54, 28	sselpwd 5256	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ∈ 𝒫 𝑆)
56	55	adantr 481	. . . . . . . . . . 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 732	. . . . . . . . . . . 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 6841	. . . . . . . . . . . . . . 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 5256	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑢 ∈ 𝒫 𝐵)
61	60	adantr 481	. . . . . . . . . . . 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 485	. . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . 14 ⊢ (𝐾‘𝑢) ∈ V
64	63	elsn 4570	. . . . . . . . . . . . 13 ⊢ ((𝐾‘𝑢) ∈ {𝐵} ↔ (𝐾‘𝑢) = 𝐵)
65	62, 64	sylibr 235	. . . . . . . . . . . 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 7000	. . . . . . . . . . 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 4129	. . . . . . . . . 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 7178	. . . . . . . . 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 12872	. . . . . . . . 9 ⊢ (((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) ⊆ (ℤ_≥‘0) ∧ (♯‘𝑢) ∈ (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) → inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ≤ (♯‘𝑢))
70	47, 68, 69	syl2an2r 691	. . . . . . . 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 9169	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → 𝑠 ∈ Fin)
72	71	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑠 ∈ Fin)
73		simplr 774	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑢 ⊊ 𝑠)
74		hashpss 32901	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ Fin ∧ 𝑢 ⊊ 𝑠) → (♯‘𝑢) < (♯‘𝑠))
75	72, 73, 74	syl2anc 590	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → (♯‘𝑢) < (♯‘𝑠))
76		simpllr 781	. . . . . . . . . 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 5098	. . . . . . . . 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 481	. . . . . . . . . . . . 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 9169	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) ∧ (𝐾‘𝑢) = 𝐵) → 𝑢 ∈ Fin)
80		hashcl 14309	. . . . . . . . . . . 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 12490	. . . . . . . . . 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 12490	. . . . . . . . . . 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 2840	. . . . . . . . . 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 11284	. . . . . . . . 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 233	. . . . . . . 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 822	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → ¬ (𝐾‘𝑢) = 𝐵)
89	88	neqned 2941	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → (𝐾‘𝑢) ≠ 𝐵)
90		df-pss 3903	. . . . . 6 ⊢ ((𝐾‘𝑢) ⊊ 𝐵 ↔ ((𝐾‘𝑢) ⊆ 𝐵 ∧ (𝐾‘𝑢) ≠ 𝐵))
91	32, 89, 90	sylanbrc 589	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) ∧ 𝑢 ⊊ 𝑠) → (𝐾‘𝑢) ⊊ 𝐵)
92	91	ex 413	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → (𝑢 ⊊ 𝑠 → (𝐾‘𝑢) ⊊ 𝐵))
93	92	alrimiv 1934	. . 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 21148	. . . 4 ⊢ (𝑊 ∈ LVec → (𝑠 ∈ 𝐽 ↔ (𝑠 ⊆ 𝐵 ∧ (𝐾‘𝑠) = 𝐵 ∧ ∀𝑢(𝑢 ⊊ 𝑠 → (𝐾‘𝑢) ⊊ 𝐵))))
96	95	biimpar 478	. . 3 ⊢ ((𝑊 ∈ LVec ∧ (𝑠 ⊆ 𝐵 ∧ (𝐾‘𝑠) = 𝐵 ∧ ∀𝑢(𝑢 ⊊ 𝑠 → (𝐾‘𝑢) ⊊ 𝐵))) → 𝑠 ∈ 𝐽)
97	3, 10, 22, 93, 96	syl13anc 1380	. 2 ⊢ (((𝜑 ∧ 𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))) ∧ (♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < )) → 𝑠 ∈ 𝐽)
98	36	elexd 3454	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ V)
99		pwidg 4549	. . . . . . . 8 ⊢ (𝑆 ∈ Fin → 𝑆 ∈ 𝒫 𝑆)
100	36, 99	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝑆)
101	36, 8	elpwd 4535	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
102		exsslsb.2	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝑆) = 𝐵)
103		fvex 6840	. . . . . . . . . 10 ⊢ (𝐾‘𝑆) ∈ V
104	103	elsn 4570	. . . . . . . . 9 ⊢ ((𝐾‘𝑆) ∈ {𝐵} ↔ (𝐾‘𝑆) = 𝐵)
105	102, 104	sylibr 235	. . . . . . . 8 ⊢ (𝜑 → (𝐾‘𝑆) ∈ {𝐵})
106	17, 101, 105	elpreimad 7000	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (^◡𝐾 “ {𝐵}))
107	100, 106	elind 4129	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))
108	49, 98, 107	fnfvimad 7178	. . . . 5 ⊢ (𝜑 → (♯‘𝑆) ∈ (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))))
109	108	ne0d 4270	. . . 4 ⊢ (𝜑 → (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))) ≠ ∅)
110		infssuzcl 12873	. . . 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 590	. . 3 ⊢ (𝜑 → inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ) ∈ (♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))))
112		fvelima2 6879	. . 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 590	. 2 ⊢ (𝜑 → ∃𝑠 ∈ (V ∩ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵})))(♯‘𝑠) = inf((♯ “ (𝒫 𝑆 ∩ (^◡𝐾 “ {𝐵}))), ℝ, < ))
114	1, 97, 7, 113	reximd2a 3249	1 ⊢ (𝜑 → ∃𝑠 ∈ 𝐽 𝑠 ⊆ 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1092 ∀wal 1545 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∃wrex 3063 Vcvv 3431 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ⊊ wpss 3884 ∅c0 4261 𝒫 cpw 4529 {csn 4555 class class class wbr 5072 ^◡ccnv 5617 “ cima 5621 Fun wfun 6479 Fn wfn 6480 ⟶wf 6481 ‘cfv 6485 Fincfn 8883 infcinf 9344 ℝcr 11028 0cc0 11029 +∞cpnf 11167 < clt 11170 ≤ cle 11171 ℕ₀cn0 12428 ℤ_≥cuz 12779 ♯chash 14283 Basecbs 17170 LModclmod 20850 LSubSpclss 20921 LSpanclspn 20961 LBasisclbs 21064 LVecclvec 21092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-xnn0 12502 df-z 12516 df-uz 12780 df-fz 13453 df-hash 14284 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904 df-sbg 18905 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-drng 20703 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lbs 21065 df-lvec 21093

This theorem is referenced by: lbslelsp 33782

W3C validator