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Theorem f1lindf 20738

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 2736	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
2	1	lindff 20731	. . . . . 6 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
3	2	ancoms 462	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
4	3	3adant3 1134	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐹:dom 𝐹⟶(Base‘𝑊))
5		f1f 6593	. . . . 5 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → 𝐺:𝐾⟶dom 𝐹)
6	5	3ad2ant3 1137	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺:𝐾⟶dom 𝐹)
7		fco 6547	. . . 4 ⊢ ((𝐹:dom 𝐹⟶(Base‘𝑊) ∧ 𝐺:𝐾⟶dom 𝐹) → (𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊))
8	4, 6, 7	syl2anc 587	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊))
9	8	ffdmd 6554	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊))
10		simpl2 1194	. . . . 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 484	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝐺:𝐾⟶dom 𝐹)
12	8	fdmd 6534	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → dom (𝐹 ∘ 𝐺) = 𝐾)
13	12	eleq2d 2816	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝑥 ∈ dom (𝐹 ∘ 𝐺) ↔ 𝑥 ∈ 𝐾))
14	13	biimpa 480	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝑥 ∈ 𝐾)
15	11, 14	ffvelrnd 6883	. . . . . 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 4027	. . . . . 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 4689	. . . . . 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 2736	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
22		eqid 2736	. . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
23		eqid 2736	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
24		eqid 2736	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
25		eqid 2736	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
26	21, 22, 23, 24, 25	lindfind 20732	. . . . 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 839	. . . 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 6594	. . . . . . . . . . 11 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → 𝐺 Fn 𝐾)
29	28	3ad2ant3 1137	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺 Fn 𝐾)
30	29	adantr 484	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝐺 Fn 𝐾)
31		fvco2 6786	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
32	30, 14, 31	syl2anc 587	. . . . . . . 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 7207	. . . . . . 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 2815	. . . . . 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 1193	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → 𝑊 ∈ LMod)
36		imassrn 5925	. . . . . . . . . . 11 ⊢ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ ran 𝐹
37	4	frnd 6531	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → ran 𝐹 ⊆ (Base‘𝑊))
38	36, 37	sstrid 3898	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ (Base‘𝑊))
39	38	adantr 484	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ (Base‘𝑊))
40		imaco 6095	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = (𝐹 “ (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})))
41	12	difeq1d 4022	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (dom (𝐹 ∘ 𝐺) ∖ {𝑥}) = (𝐾 ∖ {𝑥}))
42	41	imaeq2d 5914	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = (𝐺 “ (𝐾 ∖ {𝑥})))
43		df-f1 6363	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:𝐾–1-1→dom 𝐹 ↔ (𝐺:𝐾⟶dom 𝐹 ∧ Fun ^◡𝐺))
44	43	simprbi 500	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → Fun ^◡𝐺)
45	44	3ad2ant3 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → Fun ^◡𝐺)
46		imadif 6442	. . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})))
49	48	adantr 484	. . . . . . . . . . . 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 6768	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → {(𝐺‘𝑥)} = (𝐺 “ {𝑥}))
51	29, 50	sylan 583	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → {(𝐺‘𝑥)} = (𝐺 “ {𝑥}))
52	51	difeq2d 4023	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐺 “ 𝐾) ∖ {(𝐺‘𝑥)}) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})))
53		imassrn 5925	. . . . . . . . . . . . . . 15 ⊢ (𝐺 “ 𝐾) ⊆ ran 𝐺
54	6	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → 𝐺:𝐾⟶dom 𝐹)
55	54	frnd 6531	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ran 𝐺 ⊆ dom 𝐹)
56	53, 55	sstrid 3898	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐺 “ 𝐾) ⊆ dom 𝐹)
57	56	ssdifd 4041	. . . . . . . . . . . . 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 3926	. . . . . . . . . . . 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 3925	. . . . . . . . . . 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 5950	. . . . . . . . . . 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 3935	. . . . . . . . 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 19975	. . . . . . . . 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 594	. . . . . . 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 3886	. . . . . 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 243	. . . . 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 201	. . 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 3102	. 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 1138	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝑊 ∈ LMod)
72		rellindf 20724	. . . . . 6 ⊢ Rel LIndF
73	72	brrelex1i 5590	. . . . 5 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
74	73	3ad2ant2 1136	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐹 ∈ V)
75		simp3 1140	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺:𝐾–1-1→dom 𝐹)
76	74	dmexd 7661	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → dom 𝐹 ∈ V)
77		f1dmex 7708	. . . . . 6 ⊢ ((𝐺:𝐾–1-1→dom 𝐹 ∧ dom 𝐹 ∈ V) → 𝐾 ∈ V)
78	75, 76, 77	syl2anc 587	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐾 ∈ V)
79	6, 78	fexd 7021	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺 ∈ V)
80		coexg 7685	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V)
81	74, 79, 80	syl2anc 587	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) ∈ V)
82	1, 21, 22, 23, 25, 24	islindf 20728	. . 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 587	. 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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 Vcvv 3398 ∖ cdif 3850 ⊆ wss 3853 {csn 4527 class class class wbr 5039 ^◡ccnv 5535 dom cdm 5536 ran crn 5537 “ cima 5539 ∘ ccom 5540 Fun wfun 6352 Fn wfn 6353 ⟶wf 6354 –1-1→wf1 6355 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Scalarcsca 16752 ·_𝑠 cvsca 16753 0_gc0g 16898 LModclmod 19853 LSpanclspn 19962 LIndF clindf 20720

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 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-1cn 10752 ax-addcl 10754

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-nn 11796 df-ndx 16669 df-slot 16670 df-base 16672 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-lmod 19855 df-lss 19923 df-lsp 19963 df-lindf 20722

This theorem is referenced by: lindfres 20739 f1linds 20741

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