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Theorem lindfmm 21870

Description: Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

**Hypotheses**
Ref	Expression
lindfmm.b	⊢ 𝐵 = (Base‘𝑆)
lindfmm.c	⊢ 𝐶 = (Base‘𝑇)

**Assertion**
Ref	Expression
lindfmm	⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇))

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

Step	Hyp	Ref	Expression
1		rellindf 21851	. . . . 5 ⊢ Rel LIndF
2	1	brrelex1i 5756	. . . 4 ⊢ (𝐹 LIndF 𝑆 → 𝐹 ∈ V)
3		simp3 1138	. . . 4 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → 𝐹:𝐼⟶𝐵)
4		dmfex 7945	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐼⟶𝐵) → 𝐼 ∈ V)
5	2, 3, 4	syl2anr 596	. . 3 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝐹 LIndF 𝑆) → 𝐼 ∈ V)
6	5	ex 412	. 2 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 → 𝐼 ∈ V))
7	1	brrelex1i 5756	. . . 4 ⊢ ((𝐺 ∘ 𝐹) LIndF 𝑇 → (𝐺 ∘ 𝐹) ∈ V)
8		f1f 6817	. . . . . 6 ⊢ (𝐺:𝐵–1-1→𝐶 → 𝐺:𝐵⟶𝐶)
9		fco 6771	. . . . . 6 ⊢ ((𝐺:𝐵⟶𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐺 ∘ 𝐹):𝐼⟶𝐶)
10	8, 9	sylan 579	. . . . 5 ⊢ ((𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐺 ∘ 𝐹):𝐼⟶𝐶)
11	10	3adant1 1130	. . . 4 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐺 ∘ 𝐹):𝐼⟶𝐶)
12		dmfex 7945	. . . 4 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ (𝐺 ∘ 𝐹):𝐼⟶𝐶) → 𝐼 ∈ V)
13	7, 11, 12	syl2anr 596	. . 3 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) ∧ (𝐺 ∘ 𝐹) LIndF 𝑇) → 𝐼 ∈ V)
14	13	ex 412	. 2 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → ((𝐺 ∘ 𝐹) LIndF 𝑇 → 𝐼 ∈ V))
15		eldifi 4154	. . . . . . . . 9 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
16		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺:𝐵–1-1→𝐶)
17		lmhmlmod1 21055	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
18	17	ad3antrrr 729	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑆 ∈ LMod)
19		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
20		simprl 770	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝐹:𝐼⟶𝐵)
21		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆))) → 𝑥 ∈ 𝐼)
22		ffvelcdm 7115	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ 𝐵)
23	20, 21, 22	syl2an 595	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐹‘𝑥) ∈ 𝐵)
24		lindfmm.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝑆)
25		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
26		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆)
27		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
28	24, 25, 26, 27	lmodvscl 20898	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹‘𝑥) ∈ 𝐵) → (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ 𝐵)
29	18, 19, 23, 28	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ 𝐵)
30		imassrn 6100	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝐹
31		frn 6754	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐼⟶𝐵 → ran 𝐹 ⊆ 𝐵)
32	31	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V) → ran 𝐹 ⊆ 𝐵)
33	30, 32	sstrid 4020	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
34	33	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
35		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (LSpan‘𝑆) = (LSpan‘𝑆)
36	24, 35	lspssv 21004	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ LMod ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
37	18, 34, 36	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
38		f1elima 7300	. . . . . . . . . . . . 13 ⊢ ((𝐺:𝐵–1-1→𝐶 ∧ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ 𝐵 ∧ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵) → ((𝐺‘(𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥))) ∈ (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) ↔ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
39	16, 29, 37, 38	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺‘(𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥))) ∈ (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) ↔ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
40		simplll 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺 ∈ (𝑆 LMHom 𝑇))
41		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ ( ·_𝑠 ‘𝑇) = ( ·_𝑠 ‘𝑇)
42	25, 27, 24, 26, 41	lmhmlin 21057	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹‘𝑥) ∈ 𝐵) → (𝐺‘(𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥))) = (𝑘( ·_𝑠 ‘𝑇)(𝐺‘(𝐹‘𝑥))))
43	40, 19, 23, 42	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺‘(𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥))) = (𝑘( ·_𝑠 ‘𝑇)(𝐺‘(𝐹‘𝑥))))
44		ffn 6747	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼)
45	44	ad2antrl 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝐹 Fn 𝐼)
46		fvco2 7019	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
47	45, 21, 46	syl2an 595	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
48	47	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) = (𝑘( ·_𝑠 ‘𝑇)(𝐺‘(𝐹‘𝑥))))
49	43, 48	eqtr4d 2783	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺‘(𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥))) = (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)))
50		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (LSpan‘𝑇) = (LSpan‘𝑇)
51	24, 35, 50	lmhmlsp 21071	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))))
52	40, 34, 51	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))))
53		imaco 6282	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})) = (𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))
54	53	fveq2i 6923	. . . . . . . . . . . . . 14 ⊢ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥}))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥}))))
55	52, 54	eqtr4di 2798	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥}))))
56	49, 55	eleq12d 2838	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺‘(𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥))) ∈ (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) ↔ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
57	39, 56	bitr3d 281	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
58	57	notbid 318	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ (𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (¬ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ¬ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
59	58	anassrs 467	. . . . . . . . 9 ⊢ (((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆))) → (¬ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ¬ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
60	15, 59	sylan2 592	. . . . . . . 8 ⊢ (((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))})) → (¬ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ¬ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
61	60	ralbidva 3182	. . . . . . 7 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ 𝑥 ∈ 𝐼) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}) ¬ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}) ¬ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
62		eqid 2740	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
63	25, 62	lmhmsca 21052	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
64	63	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
65	63	fveq2d 6924	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → (0_g‘(Scalar‘𝑇)) = (0_g‘(Scalar‘𝑆)))
66	65	sneqd 4660	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → {(0_g‘(Scalar‘𝑇))} = {(0_g‘(Scalar‘𝑆))})
67	64, 66	difeq12d 4150	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑇)) ∖ {(0_g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}))
68	67	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ 𝑥 ∈ 𝐼) → ((Base‘(Scalar‘𝑇)) ∖ {(0_g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}))
69	68	raleqdv 3334	. . . . . . 7 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ 𝑥 ∈ 𝐼) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0_g‘(Scalar‘𝑇))}) ¬ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}) ¬ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
70	61, 69	bitr4d 282	. . . . . 6 ⊢ ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) ∧ 𝑥 ∈ 𝐼) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}) ¬ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0_g‘(Scalar‘𝑇))}) ¬ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
71	70	ralbidva 3182	. . . . 5 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → (∀𝑥 ∈ 𝐼 ∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}) ¬ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0_g‘(Scalar‘𝑇))}) ¬ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
72	17	ad2antrr 725	. . . . . 6 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝑆 ∈ LMod)
73		simprr 772	. . . . . 6 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝐼 ∈ V)
74		eqid 2740	. . . . . . 7 ⊢ (0_g‘(Scalar‘𝑆)) = (0_g‘(Scalar‘𝑆))
75	24, 26, 35, 25, 27, 74	islindf2 21857	. . . . . 6 ⊢ ((𝑆 ∈ LMod ∧ 𝐼 ∈ V ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}) ¬ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
76	72, 73, 20, 75	syl3anc 1371	. . . . 5 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → (𝐹 LIndF 𝑆 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}) ¬ (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
77		lmhmlmod2 21054	. . . . . . 7 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
78	77	ad2antrr 725	. . . . . 6 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝑇 ∈ LMod)
79	10	ad2ant2lr 747	. . . . . 6 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → (𝐺 ∘ 𝐹):𝐼⟶𝐶)
80		lindfmm.c	. . . . . . 7 ⊢ 𝐶 = (Base‘𝑇)
81		eqid 2740	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
82		eqid 2740	. . . . . . 7 ⊢ (0_g‘(Scalar‘𝑇)) = (0_g‘(Scalar‘𝑇))
83	80, 41, 50, 62, 81, 82	islindf2 21857	. . . . . 6 ⊢ ((𝑇 ∈ LMod ∧ 𝐼 ∈ V ∧ (𝐺 ∘ 𝐹):𝐼⟶𝐶) → ((𝐺 ∘ 𝐹) LIndF 𝑇 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0_g‘(Scalar‘𝑇))}) ¬ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
84	78, 73, 79, 83	syl3anc 1371	. . . . 5 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → ((𝐺 ∘ 𝐹) LIndF 𝑇 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0_g‘(Scalar‘𝑇))}) ¬ (𝑘( ·_𝑠 ‘𝑇)((𝐺 ∘ 𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})))))
85	71, 76, 84	3bitr4d 311	. . . 4 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇))
86	85	exp32 420	. . 3 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) → (𝐹:𝐼⟶𝐵 → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇))))
87	86	3impia 1117	. 2 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇)))
88	6, 14, 87	pm5.21ndd 379	1 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ∖ cdif 3973 ⊆ wss 3976 {csn 4648 class class class wbr 5166 ran crn 5701 “ cima 5703 ∘ ccom 5704 Fn wfn 6568 ⟶wf 6569 –1-1→wf1 6570 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 Scalarcsca 17314 ·_𝑠 cvsca 17315 0_gc0g 17499 LModclmod 20880 LSpanclspn 20992 LMHom clmhm 21041 LIndF clindf 21847

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-0g 17501 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-ghm 19253 df-mgp 20162 df-ur 20209 df-ring 20262 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lmhm 21044 df-lindf 21849

This theorem is referenced by: lindsmm 21871

W3C validator