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Theorem lindfmm 22040

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 22021	. . . . 5 ⊢ Rel LIndF
2	1	brrelex1i 5711	. . . 4 ⊢ (𝐹 LIndF 𝑆 → 𝐹 ∈ V)
3		simp3 1156	. . . 4 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → 𝐹:𝐼⟶𝐵)
4		dmfex 7902	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐹:𝐼⟶𝐵) → 𝐼 ∈ V)
5	2, 3, 4	syl2anr 609	. . 3 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) ∧ 𝐹 LIndF 𝑆) → 𝐼 ∈ V)
6	5	ex 418	. 2 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 → 𝐼 ∈ V))
7	1	brrelex1i 5711	. . . 4 ⊢ ((𝐺 ∘ 𝐹) LIndF 𝑇 → (𝐺 ∘ 𝐹) ∈ V)
8		f1f 6771	. . . . . 6 ⊢ (𝐺:𝐵–1-1→𝐶 → 𝐺:𝐵⟶𝐶)
9		fco 6727	. . . . . 6 ⊢ ((𝐺:𝐵⟶𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐺 ∘ 𝐹):𝐼⟶𝐶)
10	8, 9	sylan 592	. . . . 5 ⊢ ((𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐺 ∘ 𝐹):𝐼⟶𝐶)
11	10	3adant1 1148	. . . 4 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐺 ∘ 𝐹):𝐼⟶𝐶)
12		dmfex 7902	. . . 4 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ (𝐺 ∘ 𝐹):𝐼⟶𝐶) → 𝐼 ∈ V)
13	7, 11, 12	syl2anr 609	. . 3 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) ∧ (𝐺 ∘ 𝐹) LIndF 𝑇) → 𝐼 ∈ V)
14	13	ex 418	. 2 ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → ((𝐺 ∘ 𝐹) LIndF 𝑇 → 𝐼 ∈ V))
15		eldifi 4078	. . . . . . . . 9 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
16		simpllr 788	. . . . . . . . . . . . 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 21217	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
18	17	ad3antrrr 743	. . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝐹:𝐼⟶𝐵)
21		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆))) → 𝑥 ∈ 𝐼)
22		ffvelcdm 7074	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ 𝐵)
23	20, 21, 22	syl2an 608	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
26		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ ( ·_𝑠 ‘𝑆) = ( ·_𝑠 ‘𝑆)
27		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
28	24, 25, 26, 27	lmodvscl 21062	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹‘𝑥) ∈ 𝐵) → (𝑘( ·_𝑠 ‘𝑆)(𝐹‘𝑥)) ∈ 𝐵)
29	18, 19, 23, 28	syl3anc 1398	. . . . . . . . . . . . 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 6067	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝐹
31		frn 6710	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐼⟶𝐵 → ran 𝐹 ⊆ 𝐵)
32	31	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V) → ran 𝐹 ⊆ 𝐵)
33	30, 32	sstrid 3942	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
34	33	ad2antlr 740	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . 15 ⊢ (LSpan‘𝑆) = (LSpan‘𝑆)
36	24, 35	lspssv 21167	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ LMod ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
37	18, 34, 36	syl2anc 596	. . . . . . . . . . . . 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 7260	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 787	. . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . 16 ⊢ ( ·_𝑠 ‘𝑇) = ( ·_𝑠 ‘𝑇)
42	25, 27, 24, 26, 41	lmhmlin 21219	. . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 6702	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐼⟶𝐵 → 𝐹 Fn 𝐼)
45	44	ad2antrl 741	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝐹 Fn 𝐼)
46		fvco2 6975	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝐼 ∧ 𝑥 ∈ 𝐼) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
47	45, 21, 46	syl2an 608	. . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . 16 ⊢ (LSpan‘𝑇) = (LSpan‘𝑇)
51	24, 35, 50	lmhmlsp 21233	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 6247	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥})) = (𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))
54	53	fveq2i 6881	. . . . . . . . . . . . . 14 ⊢ ((LSpan‘𝑇)‘((𝐺 ∘ 𝐹) “ (𝐼 ∖ {𝑥}))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥}))))
55	52, 54	eqtr4di 2813	. . . . . . . . . . . . 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 2854	. . . . . . . . . . . 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 284	. . . . . . . . . . 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 321	. . . . . . . . . 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 473	. . . . . . . . 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 605	. . . . . . . 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 3183	. . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
63	25, 62	lmhmsca 21214	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
64	63	fveq2d 6882	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
65	63	fveq2d 6882	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → (0_g‘(Scalar‘𝑇)) = (0_g‘(Scalar‘𝑆)))
66	65	sneqd 4596	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → {(0_g‘(Scalar‘𝑇))} = {(0_g‘(Scalar‘𝑆))})
67	64, 66	difeq12d 4075	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑇)) ∖ {(0_g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0_g‘(Scalar‘𝑆))}))
68	67	ad3antrrr 743	. . . . . . . 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 3319	. . . . . . 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 285	. . . . . 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 3183	. . . . 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 739	. . . . . 6 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝑆 ∈ LMod)
73		simprr 785	. . . . . 6 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝐼 ∈ V)
74		eqid 2760	. . . . . . 7 ⊢ (0_g‘(Scalar‘𝑆)) = (0_g‘(Scalar‘𝑆))
75	24, 26, 35, 25, 27, 74	islindf2 22027	. . . . . 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 1398	. . . . 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 21216	. . . . . . 7 ⊢ (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
78	77	ad2antrr 739	. . . . . 6 ⊢ (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶) ∧ (𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ V)) → 𝑇 ∈ LMod)
79	10	ad2ant2lr 761	. . . . . 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 2760	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
82		eqid 2760	. . . . . . 7 ⊢ (0_g‘(Scalar‘𝑇)) = (0_g‘(Scalar‘𝑇))
83	80, 41, 50, 62, 81, 82	islindf2 22027	. . . . . 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 1398	. . . . 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 314	. . . 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 426	. . 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 1135	. 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 382	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 Vcvv 3450 ∖ cdif 3896 ⊆ wss 3899 {csn 4584 class class class wbr 5103 ran crn 5656 “ cima 5658 ∘ ccom 5659 Fn wfn 6528 ⟶wf 6529 –1-1→wf1 6530 ‘cfv 6533 (class class class)co 7413 Basecbs 17301 Scalarcsca 17345 ·_𝑠 cvsca 17346 0_gc0g 17524 LModclmod 21044 LSpanclspn 21155 LMHom clmhm 21203 LIndF clindf 22017

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-0g 17526 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-grp 19060 df-minusg 19061 df-sbg 19062 df-subg 19246 df-ghm 19341 df-mgp 20274 df-ur 20321 df-ring 20374 df-lmod 21046 df-lss 21116 df-lsp 21156 df-lmhm 21206 df-lindf 22019

This theorem is used by: lindsmm 22041

W3C validator