f1lindf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > f1lindf		Structured version Visualization version GIF version

Theorem f1lindf 21773

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 2725	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
2	1	lindff 21766	. . . . . 6 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
3	2	ancoms 457	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
4	3	3adant3 1129	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐹:dom 𝐹⟶(Base‘𝑊))
5		f1f 6793	. . . . 5 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → 𝐺:𝐾⟶dom 𝐹)
6	5	3ad2ant3 1132	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺:𝐾⟶dom 𝐹)
7		fco 6747	. . . 4 ⊢ ((𝐹:dom 𝐹⟶(Base‘𝑊) ∧ 𝐺:𝐾⟶dom 𝐹) → (𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊))
8	4, 6, 7	syl2anc 582	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊))
9	8	ffdmd 6754	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊))
10		simpl2 1189	. . . . 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 479	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝐺:𝐾⟶dom 𝐹)
12	8	fdmd 6733	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → dom (𝐹 ∘ 𝐺) = 𝐾)
13	12	eleq2d 2811	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝑥 ∈ dom (𝐹 ∘ 𝐺) ↔ 𝑥 ∈ 𝐾))
14	13	biimpa 475	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝑥 ∈ 𝐾)
15	11, 14	ffvelcdmd 7094	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → (𝐺‘𝑥) ∈ dom 𝐹)
16	15	adantrr 715	. . . . 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 4123	. . . . . 6 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
18	17	ad2antll 727	. . . . 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 4795	. . . . . 6 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
20	19	ad2antll 727	. . . . 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 2725	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
22		eqid 2725	. . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
23		eqid 2725	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
24		eqid 2725	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
25		eqid 2725	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
26	21, 22, 23, 24, 25	lindfind 21767	. . . . 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 837	. . . 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 6794	. . . . . . . . . . 11 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → 𝐺 Fn 𝐾)
29	28	3ad2ant3 1132	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺 Fn 𝐾)
30	29	adantr 479	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝐺 Fn 𝐾)
31		fvco2 6994	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
32	30, 14, 31	syl2anc 582	. . . . . . . 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 7435	. . . . . . 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 2810	. . . . . 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 1188	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → 𝑊 ∈ LMod)
36		imassrn 6075	. . . . . . . . . . 11 ⊢ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ ran 𝐹
37	4	frnd 6731	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → ran 𝐹 ⊆ (Base‘𝑊))
38	36, 37	sstrid 3988	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ (Base‘𝑊))
39	38	adantr 479	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ (Base‘𝑊))
40		imaco 6257	. . . . . . . . . 10 ⊢ ((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = (𝐹 “ (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})))
41	12	difeq1d 4117	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (dom (𝐹 ∘ 𝐺) ∖ {𝑥}) = (𝐾 ∖ {𝑥}))
42	41	imaeq2d 6064	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = (𝐺 “ (𝐾 ∖ {𝑥})))
43		df-f1 6554	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺:𝐾–1-1→dom 𝐹 ↔ (𝐺:𝐾⟶dom 𝐹 ∧ Fun ^◡𝐺))
44	43	simprbi 495	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐾–1-1→dom 𝐹 → Fun ^◡𝐺)
45	44	3ad2ant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → Fun ^◡𝐺)
46		imadif 6638	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})))
49	48	adantr 479	. . . . . . . . . . . 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 6976	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → {(𝐺‘𝑥)} = (𝐺 “ {𝑥}))
51	29, 50	sylan 578	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → {(𝐺‘𝑥)} = (𝐺 “ {𝑥}))
52	51	difeq2d 4118	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐺 “ 𝐾) ∖ {(𝐺‘𝑥)}) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})))
53		imassrn 6075	. . . . . . . . . . . . . . 15 ⊢ (𝐺 “ 𝐾) ⊆ ran 𝐺
54	6	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → 𝐺:𝐾⟶dom 𝐹)
55	54	frnd 6731	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ran 𝐺 ⊆ dom 𝐹)
56	53, 55	sstrid 3988	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐺 “ 𝐾) ⊆ dom 𝐹)
57	56	ssdifd 4137	. . . . . . . . . . . . 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 4016	. . . . . . . . . . . 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 4015	. . . . . . . . . . 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 6107	. . . . . . . . . . 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 4025	. . . . . . . . 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 20880	. . . . . . . . 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 1368	. . . . . . . 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 589	. . . . . . 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 3975	. . . . . 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 715	. . . 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 3190	. 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 1133	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝑊 ∈ LMod)
72		rellindf 21759	. . . . . 6 ⊢ Rel LIndF
73	72	brrelex1i 5734	. . . . 5 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
74	73	3ad2ant2 1131	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐹 ∈ V)
75		simp3 1135	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺:𝐾–1-1→dom 𝐹)
76	74	dmexd 7911	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → dom 𝐹 ∈ V)
77		f1dmex 7961	. . . . . 6 ⊢ ((𝐺:𝐾–1-1→dom 𝐹 ∧ dom 𝐹 ∈ V) → 𝐾 ∈ V)
78	75, 76, 77	syl2anc 582	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐾 ∈ V)
79	6, 78	fexd 7239	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺 ∈ V)
80		coexg 7937	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V)
81	74, 79, 80	syl2anc 582	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) ∈ V)
82	1, 21, 22, 23, 25, 24	islindf 21763	. . 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 582	. 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 711	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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 Vcvv 3461 ∖ cdif 3941 ⊆ wss 3944 {csn 4630 class class class wbr 5149 ^◡ccnv 5677 dom cdm 5678 ran crn 5679 “ cima 5681 ∘ ccom 5682 Fun wfun 6543 Fn wfn 6544 ⟶wf 6545 –1-1→wf1 6546 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Scalarcsca 17239 ·_𝑠 cvsca 17240 0_gc0g 17424 LModclmod 20755 LSpanclspn 20867 LIndF clindf 21755

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-1cn 11198 ax-addcl 11200

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-nn 12246 df-slot 17154 df-ndx 17166 df-base 17184 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lindf 21757

This theorem is referenced by: lindfres 21774 f1linds 21776

W3C validator