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Theorem lindfmm 20965

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 20946	. . . . 5 ⊢ Rel LIndF
2	1	brrelex1i 5602	. . . 4 ⊢ (𝐹 LIndF 𝑆 → 𝐹 ∈ V)
3		simp3 1134	. . . 4 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → 𝐹:𝐼⟶𝐵)
4		dmfex 7635	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐼⟶𝐵) → 𝐼 ∈ V)
5	2, 3, 4	syl2anr 598	. . 3 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝐹 LIndF 𝑆) → 𝐼 ∈ V)
6	5	ex 415	. 2 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 → 𝐼 ∈ V))
7	1	brrelex1i 5602	. . . 4 ⊢ ((𝐺 ∘ 𝐹) LIndF 𝑇 → (𝐺 ∘ 𝐹) ∈ V)
8		f1f 6569	. . . . . 6 ⊢ (𝐺:𝐵–1-1→𝐶 → 𝐺:𝐵⟶𝐶)
9		fco 6525	. . . . . 6 ⊢ ((𝐺:𝐵⟶𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐺 ∘ 𝐹):𝐼⟶𝐶)
10	8, 9	sylan 582	. . . . 5 ⊢ ((𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐺 ∘ 𝐹):𝐼⟶𝐶)
11	10	3adant1 1126	. . . 4 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐺 ∘ 𝐹):𝐼⟶𝐶)
12		dmfex 7635	. . . 4 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ (𝐺 ∘ 𝐹):𝐼⟶𝐶) → 𝐼 ∈ V)
13	7, 11, 12	syl2anr 598	. . 3 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) ∧ (𝐺 ∘ 𝐹) LIndF 𝑇) → 𝐼 ∈ V)
14	13	ex 415	. 2 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → ((𝐺 ∘ 𝐹) LIndF 𝑇 → 𝐼 ∈ V))
15		eldifi 4102	. . . . . . . . 9 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
16		simpllr 774	. . . . . . . . . . . . 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 19799	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
18	17	ad3antrrr 728	. . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝐹:𝐼⟶𝐵)
21		simpl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆))) → 𝑥 ∈ 𝐼)
22		ffvelrn 6843	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ 𝐵)
23	20, 21, 22	syl2an 597	. . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
26		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆)
27		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
28	24, 25, 26, 27	lmodvscl 19645	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹‘𝑥) ∈ 𝐵) → (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ 𝐵)
29	18, 19, 23, 28	syl3anc 1367	. . . . . . . . . . . . 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 5934	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝐹
31		frn 6514	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐼⟶𝐵 → ran 𝐹 ⊆ 𝐵)
32	31	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V) → ran 𝐹 ⊆ 𝐵)
33	30, 32	sstrid 3977	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
34	33	ad2antlr 725	. . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . 15 ⊢ (LSpan‘𝑆) = (LSpan‘𝑆)
36	24, 35	lspssv 19749	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ LMod ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
37	18, 34, 36	syl2anc 586	. . . . . . . . . . . . 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 7015	. . . . . . . . . . . . 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 1367	. . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . 16 ⊢ ( ·_𝑠 ‘𝑇) = ( ·_𝑠 ‘𝑇)
42	25, 27, 24, 26, 41	lmhmlin 19801	. . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . 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 6508	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼)
45	44	ad2antrl 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝐹 Fn 𝐼)
46		fvco2 6752	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
47	45, 21, 46	syl2an 597	. . . . . . . . . . . . . . 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 7166	. . . . . . . . . . . . . 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 2859	. . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . 16 ⊢ (LSpan‘𝑇) = (LSpan‘𝑇)
51	24, 35, 50	lmhmlsp 19815	. . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . 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 6098	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})) = (𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))
54	53	fveq2i 6667	. . . . . . . . . . . . . 14 ⊢ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥}))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥}))))
55	52, 54	syl6eqr 2874	. . . . . . . . . . . . 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 2907	. . . . . . . . . . . 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 283	. . . . . . . . . . 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 320	. . . . . . . . . 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 470	. . . . . . . . 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 594	. . . . . . . 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 3196	. . . . . . 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 2821	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
63	25, 62	lmhmsca 19796	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
64	63	fveq2d 6668	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
65	63	fveq2d 6668	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → (0_g‘(Scalar‘𝑇)) = (0_g‘(Scalar‘𝑆)))
66	65	sneqd 4572	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → {(0_g‘(Scalar‘𝑇))} = {(0_g‘(Scalar‘𝑆))})
67	64, 66	difeq12d 4099	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑇)) ∖ {(0_g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}))
68	67	ad3antrrr 728	. . . . . . . 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 3415	. . . . . . 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 284	. . . . . 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 3196	. . . . 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 724	. . . . . 6 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝑆 ∈ LMod)
73		simprr 771	. . . . . 6 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝐼 ∈ V)
74		eqid 2821	. . . . . . 7 ⊢ (0_g‘(Scalar‘𝑆)) = (0_g‘(Scalar‘𝑆))
75	24, 26, 35, 25, 27, 74	islindf2 20952	. . . . . 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 1367	. . . . 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 19798	. . . . . . 7 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
78	77	ad2antrr 724	. . . . . 6 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝑇 ∈ LMod)
79	10	ad2ant2lr 746	. . . . . 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 2821	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
82		eqid 2821	. . . . . . 7 ⊢ (0_g‘(Scalar‘𝑇)) = (0_g‘(Scalar‘𝑇))
83	80, 41, 50, 62, 81, 82	islindf2 20952	. . . . . 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 1367	. . . . 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 313	. . . 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 423	. . 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 1113	. 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 383	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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 {csn 4560 class class class wbr 5058 ran crn 5550 “ cima 5552 ∘ ccom 5553 Fn wfn 6344 ⟶wf 6345 –1-1→wf1 6346 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Scalarcsca 16562 ·_𝑠 cvsca 16563 0_gc0g 16707 LModclmod 19628 LSpanclspn 19737 LMHom clmhm 19785 LIndF clindf 20942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-ghm 18350 df-mgp 19234 df-ur 19246 df-ring 19293 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lmhm 19788 df-lindf 20944

This theorem is referenced by: lindsmm 20966

W3C validator