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Theorem lindfmm 20094
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:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))

Proof of Theorem lindfmm
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rellindf 20075 . . . . 5 Rel LIndF
21brrelexi 5123 . . . 4 (𝐹 LIndF 𝑆𝐹 ∈ V)
3 simp3 1061 . . . 4 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → 𝐹:𝐼𝐵)
4 dmfex 7078 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝐼𝐵) → 𝐼 ∈ V)
52, 3, 4syl2anr 495 . . 3 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) ∧ 𝐹 LIndF 𝑆) → 𝐼 ∈ V)
65ex 450 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆𝐼 ∈ V))
71brrelexi 5123 . . . 4 ((𝐺𝐹) LIndF 𝑇 → (𝐺𝐹) ∈ V)
8 f1f 6063 . . . . . 6 (𝐺:𝐵1-1𝐶𝐺:𝐵𝐶)
9 fco 6020 . . . . . 6 ((𝐺:𝐵𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
108, 9sylan 488 . . . . 5 ((𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
11103adant1 1077 . . . 4 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
12 dmfex 7078 . . . 4 (((𝐺𝐹) ∈ V ∧ (𝐺𝐹):𝐼𝐶) → 𝐼 ∈ V)
137, 11, 12syl2anr 495 . . 3 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) ∧ (𝐺𝐹) LIndF 𝑇) → 𝐼 ∈ V)
1413ex 450 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → ((𝐺𝐹) LIndF 𝑇𝐼 ∈ V))
15 eldifi 3715 . . . . . . . . 9 (𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
16 simpllr 798 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺:𝐵1-1𝐶)
17 lmhmlmod1 18961 . . . . . . . . . . . . . . 15 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
1817ad3antrrr 765 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑆 ∈ LMod)
19 simprr 795 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
20 simprl 793 . . . . . . . . . . . . . . 15 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐹:𝐼𝐵)
21 simpl 473 . . . . . . . . . . . . . . 15 ((𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆))) → 𝑥𝐼)
22 ffvelrn 6318 . . . . . . . . . . . . . . 15 ((𝐹:𝐼𝐵𝑥𝐼) → (𝐹𝑥) ∈ 𝐵)
2320, 21, 22syl2an 494 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐹𝑥) ∈ 𝐵)
24 lindfmm.b . . . . . . . . . . . . . . 15 𝐵 = (Base‘𝑆)
25 eqid 2621 . . . . . . . . . . . . . . 15 (Scalar‘𝑆) = (Scalar‘𝑆)
26 eqid 2621 . . . . . . . . . . . . . . 15 ( ·𝑠𝑆) = ( ·𝑠𝑆)
27 eqid 2621 . . . . . . . . . . . . . . 15 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
2824, 25, 26, 27lmodvscl 18808 . . . . . . . . . . . . . 14 ((𝑆 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑥) ∈ 𝐵) → (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵)
2918, 19, 23, 28syl3anc 1323 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵)
30 imassrn 5441 . . . . . . . . . . . . . . . 16 (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝐹
31 frn 6015 . . . . . . . . . . . . . . . . 17 (𝐹:𝐼𝐵 → ran 𝐹𝐵)
3231adantr 481 . . . . . . . . . . . . . . . 16 ((𝐹:𝐼𝐵𝐼 ∈ V) → ran 𝐹𝐵)
3330, 32syl5ss 3598 . . . . . . . . . . . . . . 15 ((𝐹:𝐼𝐵𝐼 ∈ V) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
3433ad2antlr 762 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
35 eqid 2621 . . . . . . . . . . . . . . 15 (LSpan‘𝑆) = (LSpan‘𝑆)
3624, 35lspssv 18911 . . . . . . . . . . . . . 14 ((𝑆 ∈ LMod ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
3718, 34, 36syl2anc 692 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
38 f1elima 6480 . . . . . . . . . . . . 13 ((𝐺:𝐵1-1𝐶 ∧ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵 ∧ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵) → ((𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) ∈ (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) ↔ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
3916, 29, 37, 38syl3anc 1323 . . . . . . . . . . . 12 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) ∈ (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) ↔ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
40 simplll 797 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺 ∈ (𝑆 LMHom 𝑇))
41 eqid 2621 . . . . . . . . . . . . . . . 16 ( ·𝑠𝑇) = ( ·𝑠𝑇)
4225, 27, 24, 26, 41lmhmlin 18963 . . . . . . . . . . . . . . 15 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑥) ∈ 𝐵) → (𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) = (𝑘( ·𝑠𝑇)(𝐺‘(𝐹𝑥))))
4340, 19, 23, 42syl3anc 1323 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) = (𝑘( ·𝑠𝑇)(𝐺‘(𝐹𝑥))))
44 ffn 6007 . . . . . . . . . . . . . . . . 17 (𝐹:𝐼𝐵𝐹 Fn 𝐼)
4544ad2antrl 763 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐹 Fn 𝐼)
46 fvco2 6235 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐼𝑥𝐼) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4745, 21, 46syl2an 494 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4847oveq2d 6626 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) = (𝑘( ·𝑠𝑇)(𝐺‘(𝐹𝑥))))
4943, 48eqtr4d 2658 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) = (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)))
50 eqid 2621 . . . . . . . . . . . . . . . 16 (LSpan‘𝑇) = (LSpan‘𝑇)
5124, 35, 50lmhmlsp 18977 . . . . . . . . . . . . . . 15 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))))
5240, 34, 51syl2anc 692 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))))
53 imaco 5604 . . . . . . . . . . . . . . 15 ((𝐺𝐹) “ (𝐼 ∖ {𝑥})) = (𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))
5453fveq2i 6156 . . . . . . . . . . . . . 14 ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥}))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥}))))
5552, 54syl6eqr 2673 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥}))))
5649, 55eleq12d 2692 . . . . . . . . . . . 12 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) ∈ (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) ↔ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
5739, 56bitr3d 270 . . . . . . . . . . 11 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
5857notbid 308 . . . . . . . . . 10 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
5958anassrs 679 . . . . . . . . 9 (((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆))) → (¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
6015, 59sylan2 491 . . . . . . . 8 (((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))})) → (¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
6160ralbidva 2980 . . . . . . 7 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
62 eqid 2621 . . . . . . . . . . . 12 (Scalar‘𝑇) = (Scalar‘𝑇)
6325, 62lmhmsca 18958 . . . . . . . . . . 11 (𝐺 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
6463fveq2d 6157 . . . . . . . . . 10 (𝐺 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
6563fveq2d 6157 . . . . . . . . . . 11 (𝐺 ∈ (𝑆 LMHom 𝑇) → (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑆)))
6665sneqd 4165 . . . . . . . . . 10 (𝐺 ∈ (𝑆 LMHom 𝑇) → {(0g‘(Scalar‘𝑇))} = {(0g‘(Scalar‘𝑆))})
6764, 66difeq12d 3712 . . . . . . . . 9 (𝐺 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}))
6867ad3antrrr 765 . . . . . . . 8 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) → ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}))
6968raleqdv 3136 . . . . . . 7 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
7061, 69bitr4d 271 . . . . . 6 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
7170ralbidva 2980 . . . . 5 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
7217ad2antrr 761 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝑆 ∈ LMod)
73 simprr 795 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐼 ∈ V)
74 eqid 2621 . . . . . . 7 (0g‘(Scalar‘𝑆)) = (0g‘(Scalar‘𝑆))
7524, 26, 35, 25, 27, 74islindf2 20081 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝐼 ∈ V ∧ 𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆 ↔ ∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
7672, 73, 20, 75syl3anc 1323 . . . . 5 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐹 LIndF 𝑆 ↔ ∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
77 lmhmlmod2 18960 . . . . . . 7 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7877ad2antrr 761 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝑇 ∈ LMod)
7910ad2ant2lr 783 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐺𝐹):𝐼𝐶)
80 lindfmm.c . . . . . . 7 𝐶 = (Base‘𝑇)
81 eqid 2621 . . . . . . 7 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
82 eqid 2621 . . . . . . 7 (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑇))
8380, 41, 50, 62, 81, 82islindf2 20081 . . . . . 6 ((𝑇 ∈ LMod ∧ 𝐼 ∈ V ∧ (𝐺𝐹):𝐼𝐶) → ((𝐺𝐹) LIndF 𝑇 ↔ ∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
8478, 73, 79, 83syl3anc 1323 . . . . 5 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → ((𝐺𝐹) LIndF 𝑇 ↔ ∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
8571, 76, 843bitr4d 300 . . . 4 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))
8685exp32 630 . . 3 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) → (𝐹:𝐼𝐵 → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))))
87863impia 1258 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇)))
886, 14, 87pm5.21ndd 369 1 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  cdif 3556  wss 3559  {csn 4153   class class class wbr 4618  ran crn 5080  cima 5082  ccom 5083   Fn wfn 5847  wf 5848  1-1wf1 5849  cfv 5852  (class class class)co 6610  Basecbs 15788  Scalarcsca 15872   ·𝑠 cvsca 15873  0gc0g 16028  LModclmod 18791  LSpanclspn 18899   LMHom clmhm 18947   LIndF clindf 20071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-ress 15795  df-plusg 15882  df-0g 16030  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-grp 17353  df-minusg 17354  df-sbg 17355  df-subg 17519  df-ghm 17586  df-mgp 18418  df-ur 18430  df-ring 18477  df-lmod 18793  df-lss 18861  df-lsp 18900  df-lmhm 18950  df-lindf 20073
This theorem is referenced by:  lindsmm  20095
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