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Theorem f1lindf 20939

Description: Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)

**Assertion**
Ref	Expression
f1lindf	⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) LIndF 𝑊)

Proof of Theorem f1lindf Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
2	1	lindff 20932	. . . . . 6 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
3	2	ancoms 458	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
4	3	3adant3 1130	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐹:dom 𝐹⟶(Base‘𝑊))
5		f1f 6654	. . . . 5 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → 𝐺:𝐾⟶dom 𝐹)
6	5	3ad2ant3 1133	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺:𝐾⟶dom 𝐹)
7		fco 6608	. . . 4 ⊢ ((𝐹:dom 𝐹⟶(Base‘𝑊) ∧ 𝐺:𝐾⟶dom 𝐹) → (𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊))
8	4, 6, 7	syl2anc 583	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊))
9	8	ffdmd 6615	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊))
10		simpl2 1190	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊)
11	6	adantr 480	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝐺:𝐾⟶dom 𝐹)
12	8	fdmd 6595	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → dom (𝐹 ∘ 𝐺) = 𝐾)
13	12	eleq2d 2824	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝑥 ∈ dom (𝐹 ∘ 𝐺) ↔ 𝑥 ∈ 𝐾))
14	13	biimpa 476	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝑥 ∈ 𝐾)
15	11, 14	ffvelrnd 6944	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → (𝐺‘𝑥) ∈ dom 𝐹)
16	15	adantrr 713	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → (𝐺‘𝑥) ∈ dom 𝐹)
17		eldifi 4057	. . . . . 6 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
18	17	ad2antll 725	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
19		eldifsni 4720	. . . . . 6 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
20	19	ad2antll 725	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
21		eqid 2738	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
22		eqid 2738	. . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
23		eqid 2738	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
24		eqid 2738	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
25		eqid 2738	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
26	21, 22, 23, 24, 25	lindfind 20933	. . . . 5 ⊢ (((𝐹 LIndF 𝑊 ∧ (𝐺‘𝑥) ∈ dom 𝐹) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊)))) → ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))))
27	10, 16, 18, 20, 26	syl22anc 835	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))))
28		f1fn 6655	. . . . . . . . . . 11 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → 𝐺 Fn 𝐾)
29	28	3ad2ant3 1133	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺 Fn 𝐾)
30	29	adantr 480	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝐺 Fn 𝐾)
31		fvco2 6847	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
32	30, 14, 31	syl2anc 583	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
33	32	oveq2d 7271	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → (𝑘( ·_𝑠 ‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) = (𝑘( ·_𝑠 ‘𝑊)(𝐹‘(𝐺‘𝑥))))
34	33	eleq1d 2823	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → ((𝑘( ·_𝑠 ‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ↔ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})))))
35		simpl1 1189	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → 𝑊 ∈ LMod)
36		imassrn 5969	. . . . . . . . . . 11 ⊢ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ ran 𝐹
37	4	frnd 6592	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → ran 𝐹 ⊆ (Base‘𝑊))
38	36, 37	sstrid 3928	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ (Base‘𝑊))
39	38	adantr 480	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ (Base‘𝑊))
40		imaco 6144	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = (𝐹 “ (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})))
41	12	difeq1d 4052	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (dom (𝐹 ∘ 𝐺) ∖ {𝑥}) = (𝐾 ∖ {𝑥}))
42	41	imaeq2d 5958	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = (𝐺 “ (𝐾 ∖ {𝑥})))
43		df-f1 6423	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:𝐾–1-1→dom 𝐹 ↔ (𝐺:𝐾⟶dom 𝐹 ∧ Fun ^◡𝐺))
44	43	simprbi 496	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → Fun ^◡𝐺)
45	44	3ad2ant3 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → Fun ^◡𝐺)
46		imadif 6502	. . . . . . . . . . . . . . 15 ⊢ (Fun ^◡𝐺 → (𝐺 “ (𝐾 ∖ {𝑥})) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})))
47	45, 46	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (𝐾 ∖ {𝑥})) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})))
48	42, 47	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})))
49	48	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})))
50		fnsnfv 6829	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → {(𝐺‘𝑥)} = (𝐺 “ {𝑥}))
51	29, 50	sylan 579	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → {(𝐺‘𝑥)} = (𝐺 “ {𝑥}))
52	51	difeq2d 4053	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐺 “ 𝐾) ∖ {(𝐺‘𝑥)}) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})))
53		imassrn 5969	. . . . . . . . . . . . . . 15 ⊢ (𝐺 “ 𝐾) ⊆ ran 𝐺
54	6	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → 𝐺:𝐾⟶dom 𝐹)
55	54	frnd 6592	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ran 𝐺 ⊆ dom 𝐹)
56	53, 55	sstrid 3928	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐺 “ 𝐾) ⊆ dom 𝐹)
57	56	ssdifd 4071	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐺 “ 𝐾) ∖ {(𝐺‘𝑥)}) ⊆ (dom 𝐹 ∖ {(𝐺‘𝑥)}))
58	52, 57	eqsstrrd 3956	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺‘𝑥)}))
59	49, 58	eqsstrd 3955	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺‘𝑥)}))
60		imass2 5999	. . . . . . . . . . 11 ⊢ ((𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺‘𝑥)}) → (𝐹 “ (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))
61	59, 60	syl 17	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐹 “ (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))
62	40, 61	eqsstrid 3965	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))
63	1, 22	lspss 20161	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ (Base‘𝑊) ∧ ((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))) → ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))))
64	35, 39, 62, 63	syl3anc 1369	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))))
65	14, 64	syldan 590	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))))
66	65	sseld 3916	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → ((𝑘( ·_𝑠 ‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) → (𝑘( ·_𝑠 ‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))))
67	34, 66	sylbid 239	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → ((𝑘( ·_𝑠 ‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) → (𝑘( ·_𝑠 ‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))))
68	67	adantrr 713	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ((𝑘( ·_𝑠 ‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) → (𝑘( ·_𝑠 ‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))))
69	27, 68	mtod 197	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·_𝑠 ‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))))
70	69	ralrimivva 3114	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))))
71		simp1 1134	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝑊 ∈ LMod)
72		rellindf 20925	. . . . . 6 ⊢ Rel LIndF
73	72	brrelex1i 5634	. . . . 5 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
74	73	3ad2ant2 1132	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐹 ∈ V)
75		simp3 1136	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺:𝐾–1-1→dom 𝐹)
76	74	dmexd 7726	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → dom 𝐹 ∈ V)
77		f1dmex 7773	. . . . . 6 ⊢ ((𝐺:𝐾–1-1→dom 𝐹 ∧ dom 𝐹 ∈ V) → 𝐾 ∈ V)
78	75, 76, 77	syl2anc 583	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐾 ∈ V)
79	6, 78	fexd 7085	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺 ∈ V)
80		coexg 7750	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V)
81	74, 79, 80	syl2anc 583	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) ∈ V)
82	1, 21, 22, 23, 25, 24	islindf 20929	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝐹 ∘ 𝐺) ∈ V) → ((𝐹 ∘ 𝐺) LIndF 𝑊 ↔ ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))))))
83	71, 81, 82	syl2anc 583	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → ((𝐹 ∘ 𝐺) LIndF 𝑊 ↔ ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))))))
84	9, 70, 83	mpbir2and 709	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) LIndF 𝑊)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 {csn 4558 class class class wbr 5070 ^◡ccnv 5579 dom cdm 5580 ran crn 5581 “ cima 5583 ∘ ccom 5584 Fun wfun 6412 Fn wfn 6413 ⟶wf 6414 –1-1→wf1 6415 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 Scalarcsca 16891 ·_𝑠 cvsca 16892 0_gc0g 17067 LModclmod 20038 LSpanclspn 20148 LIndF clindf 20921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-1cn 10860 ax-addcl 10862

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-nn 11904 df-slot 16811 df-ndx 16823 df-base 16841 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lindf 20923

This theorem is referenced by: lindfres 20940 f1linds 20942

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