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Theorem f1lindf 21759

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 2731	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
2	1	lindff 21752	. . . . . 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 1132	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐹:dom 𝐹⟶(Base‘𝑊))
5		f1f 6719	. . . . 5 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → 𝐺:𝐾⟶dom 𝐹)
6	5	3ad2ant3 1135	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺:𝐾⟶dom 𝐹)
7		fco 6675	. . . 4 ⊢ ((𝐹:dom 𝐹⟶(Base‘𝑊) ∧ 𝐺:𝐾⟶dom 𝐹) → (𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊))
8	4, 6, 7	syl2anc 584	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊))
9	8	ffdmd 6681	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊))
10		simpl2 1193	. . . . 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 6661	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → dom (𝐹 ∘ 𝐺) = 𝐾)
13	12	eleq2d 2817	. . . . . . . 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	ffvelcdmd 7018	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → (𝐺‘𝑥) ∈ dom 𝐹)
16	15	adantrr 717	. . . . 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 4078	. . . . . 6 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
18	17	ad2antll 729	. . . . 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 4739	. . . . . 6 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
20	19	ad2antll 729	. . . . 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 2731	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
22		eqid 2731	. . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
23		eqid 2731	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
24		eqid 2731	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
25		eqid 2731	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
26	21, 22, 23, 24, 25	lindfind 21753	. . . . 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 838	. . . 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 6720	. . . . . . . . . . 11 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → 𝐺 Fn 𝐾)
29	28	3ad2ant3 1135	. . . . . . . . . 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 6919	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
32	30, 14, 31	syl2anc 584	. . . . . . . 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 7362	. . . . . . 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 2816	. . . . . 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 1192	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → 𝑊 ∈ LMod)
36		imassrn 6019	. . . . . . . . . . 11 ⊢ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ ran 𝐹
37	4	frnd 6659	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → ran 𝐹 ⊆ (Base‘𝑊))
38	36, 37	sstrid 3941	. . . . . . . . . 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 6198	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = (𝐹 “ (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})))
41	12	difeq1d 4072	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (dom (𝐹 ∘ 𝐺) ∖ {𝑥}) = (𝐾 ∖ {𝑥}))
42	41	imaeq2d 6008	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = (𝐺 “ (𝐾 ∖ {𝑥})))
43		df-f1 6486	. . . . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → Fun ^◡𝐺)
46		imadif 6565	. . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . 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 6901	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → {(𝐺‘𝑥)} = (𝐺 “ {𝑥}))
51	29, 50	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → {(𝐺‘𝑥)} = (𝐺 “ {𝑥}))
52	51	difeq2d 4073	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐺 “ 𝐾) ∖ {(𝐺‘𝑥)}) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})))
53		imassrn 6019	. . . . . . . . . . . . . . 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 6659	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ran 𝐺 ⊆ dom 𝐹)
56	53, 55	sstrid 3941	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐺 “ 𝐾) ⊆ dom 𝐹)
57	56	ssdifd 4092	. . . . . . . . . . . . 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 3965	. . . . . . . . . . . 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 3964	. . . . . . . . . . 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 6050	. . . . . . . . . . 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 3968	. . . . . . . . 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 20917	. . . . . . . . 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 1373	. . . . . . . 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 591	. . . . . . 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 3928	. . . . . 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 240	. . . . 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 717	. . . 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 198	. . 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 3175	. 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 1136	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝑊 ∈ LMod)
72		rellindf 21745	. . . . . 6 ⊢ Rel LIndF
73	72	brrelex1i 5670	. . . . 5 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
74	73	3ad2ant2 1134	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐹 ∈ V)
75		simp3 1138	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺:𝐾–1-1→dom 𝐹)
76	74	dmexd 7833	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → dom 𝐹 ∈ V)
77		f1dmex 7889	. . . . . 6 ⊢ ((𝐺:𝐾–1-1→dom 𝐹 ∧ dom 𝐹 ∈ V) → 𝐾 ∈ V)
78	75, 76, 77	syl2anc 584	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐾 ∈ V)
79	6, 78	fexd 7161	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺 ∈ V)
80		coexg 7859	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V)
81	74, 79, 80	syl2anc 584	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) ∈ V)
82	1, 21, 22, 23, 25, 24	islindf 21749	. . 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 584	. 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 713	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 Vcvv 3436 ∖ cdif 3894 ⊆ wss 3897 {csn 4573 class class class wbr 5089 ^◡ccnv 5613 dom cdm 5614 ran crn 5615 “ cima 5617 ∘ ccom 5618 Fun wfun 6475 Fn wfn 6476 ⟶wf 6477 –1-1→wf1 6478 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 Scalarcsca 17164 ·_𝑠 cvsca 17165 0_gc0g 17343 LModclmod 20793 LSpanclspn 20904 LIndF clindf 21741

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-1cn 11064 ax-addcl 11066

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-om 7797 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-nn 12126 df-slot 17093 df-ndx 17105 df-base 17121 df-0g 17345 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-grp 18849 df-lmod 20795 df-lss 20865 df-lsp 20905 df-lindf 21743

This theorem is referenced by: lindfres 21760 f1linds 21762

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