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Theorem lindfmm 20445
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 20426 . . . . 5 Rel LIndF
21brrelex1i 5330 . . . 4 (𝐹 LIndF 𝑆𝐹 ∈ V)
3 simp3 1168 . . . 4 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → 𝐹:𝐼𝐵)
4 dmfex 7324 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝐼𝐵) → 𝐼 ∈ V)
52, 3, 4syl2anr 590 . . 3 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) ∧ 𝐹 LIndF 𝑆) → 𝐼 ∈ V)
65ex 401 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆𝐼 ∈ V))
71brrelex1i 5330 . . . 4 ((𝐺𝐹) LIndF 𝑇 → (𝐺𝐹) ∈ V)
8 f1f 6285 . . . . . 6 (𝐺:𝐵1-1𝐶𝐺:𝐵𝐶)
9 fco 6242 . . . . . 6 ((𝐺:𝐵𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
108, 9sylan 575 . . . . 5 ((𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
11103adant1 1160 . . . 4 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
12 dmfex 7324 . . . 4 (((𝐺𝐹) ∈ V ∧ (𝐺𝐹):𝐼𝐶) → 𝐼 ∈ V)
137, 11, 12syl2anr 590 . . 3 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) ∧ (𝐺𝐹) LIndF 𝑇) → 𝐼 ∈ V)
1413ex 401 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → ((𝐺𝐹) LIndF 𝑇𝐼 ∈ V))
15 eldifi 3896 . . . . . . . . 9 (𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
16 simpllr 793 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺:𝐵1-1𝐶)
17 lmhmlmod1 19308 . . . . . . . . . . . . . . 15 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
1817ad3antrrr 721 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑆 ∈ LMod)
19 simprr 789 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
20 simprl 787 . . . . . . . . . . . . . . 15 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐹:𝐼𝐵)
21 simpl 474 . . . . . . . . . . . . . . 15 ((𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆))) → 𝑥𝐼)
22 ffvelrn 6549 . . . . . . . . . . . . . . 15 ((𝐹:𝐼𝐵𝑥𝐼) → (𝐹𝑥) ∈ 𝐵)
2320, 21, 22syl2an 589 . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . . 15 (Scalar‘𝑆) = (Scalar‘𝑆)
26 eqid 2765 . . . . . . . . . . . . . . 15 ( ·𝑠𝑆) = ( ·𝑠𝑆)
27 eqid 2765 . . . . . . . . . . . . . . 15 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
2824, 25, 26, 27lmodvscl 19152 . . . . . . . . . . . . . 14 ((𝑆 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑥) ∈ 𝐵) → (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵)
2918, 19, 23, 28syl3anc 1490 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵)
30 imassrn 5661 . . . . . . . . . . . . . . . 16 (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝐹
31 frn 6231 . . . . . . . . . . . . . . . . 17 (𝐹:𝐼𝐵 → ran 𝐹𝐵)
3231adantr 472 . . . . . . . . . . . . . . . 16 ((𝐹:𝐼𝐵𝐼 ∈ V) → ran 𝐹𝐵)
3330, 32syl5ss 3774 . . . . . . . . . . . . . . 15 ((𝐹:𝐼𝐵𝐼 ∈ V) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
3433ad2antlr 718 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
35 eqid 2765 . . . . . . . . . . . . . . 15 (LSpan‘𝑆) = (LSpan‘𝑆)
3624, 35lspssv 19258 . . . . . . . . . . . . . 14 ((𝑆 ∈ LMod ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
3718, 34, 36syl2anc 579 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
38 f1elima 6714 . . . . . . . . . . . . 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 1490 . . . . . . . . . . . 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 791 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺 ∈ (𝑆 LMHom 𝑇))
41 eqid 2765 . . . . . . . . . . . . . . . 16 ( ·𝑠𝑇) = ( ·𝑠𝑇)
4225, 27, 24, 26, 41lmhmlin 19310 . . . . . . . . . . . . . . 15 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑥) ∈ 𝐵) → (𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) = (𝑘( ·𝑠𝑇)(𝐺‘(𝐹𝑥))))
4340, 19, 23, 42syl3anc 1490 . . . . . . . . . . . . . 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 6225 . . . . . . . . . . . . . . . . 17 (𝐹:𝐼𝐵𝐹 Fn 𝐼)
4544ad2antrl 719 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐹 Fn 𝐼)
46 fvco2 6464 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐼𝑥𝐼) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4745, 21, 46syl2an 589 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4847oveq2d 6860 . . . . . . . . . . . . . 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 2802 . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . . . 16 (LSpan‘𝑇) = (LSpan‘𝑇)
5124, 35, 50lmhmlsp 19324 . . . . . . . . . . . . . . 15 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))))
5240, 34, 51syl2anc 579 . . . . . . . . . . . . . 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 5828 . . . . . . . . . . . . . . 15 ((𝐺𝐹) “ (𝐼 ∖ {𝑥})) = (𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))
5453fveq2i 6380 . . . . . . . . . . . . . 14 ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥}))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥}))))
5552, 54syl6eqr 2817 . . . . . . . . . . . . 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 2838 . . . . . . . . . . . 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 272 . . . . . . . . . . 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 309 . . . . . . . . . 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 459 . . . . . . . . 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 586 . . . . . . . 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 3132 . . . . . . 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 2765 . . . . . . . . . . . 12 (Scalar‘𝑇) = (Scalar‘𝑇)
6325, 62lmhmsca 19305 . . . . . . . . . . 11 (𝐺 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
6463fveq2d 6381 . . . . . . . . . 10 (𝐺 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
6563fveq2d 6381 . . . . . . . . . . 11 (𝐺 ∈ (𝑆 LMHom 𝑇) → (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑆)))
6665sneqd 4348 . . . . . . . . . 10 (𝐺 ∈ (𝑆 LMHom 𝑇) → {(0g‘(Scalar‘𝑇))} = {(0g‘(Scalar‘𝑆))})
6764, 66difeq12d 3893 . . . . . . . . 9 (𝐺 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}))
6867ad3antrrr 721 . . . . . . . 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 3292 . . . . . . 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 273 . . . . . 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 3132 . . . . 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 717 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝑆 ∈ LMod)
73 simprr 789 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐼 ∈ V)
74 eqid 2765 . . . . . . 7 (0g‘(Scalar‘𝑆)) = (0g‘(Scalar‘𝑆))
7524, 26, 35, 25, 27, 74islindf2 20432 . . . . . 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 1490 . . . . 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 19307 . . . . . . 7 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7877ad2antrr 717 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝑇 ∈ LMod)
7910ad2ant2lr 754 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐺𝐹):𝐼𝐶)
80 lindfmm.c . . . . . . 7 𝐶 = (Base‘𝑇)
81 eqid 2765 . . . . . . 7 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
82 eqid 2765 . . . . . . 7 (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑇))
8380, 41, 50, 62, 81, 82islindf2 20432 . . . . . 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 1490 . . . . 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 302 . . . 4 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))
8685exp32 411 . . 3 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) → (𝐹:𝐼𝐵 → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))))
87863impia 1145 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇)))
886, 14, 87pm5.21ndd 370 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cdif 3731  wss 3734  {csn 4336   class class class wbr 4811  ran crn 5280  cima 5282  ccom 5283   Fn wfn 6065  wf 6066  1-1wf1 6067  cfv 6070  (class class class)co 6844  Basecbs 16133  Scalarcsca 16220   ·𝑠 cvsca 16221  0gc0g 16369  LModclmod 19135  LSpanclspn 19246   LMHom clmhm 19294   LIndF clindf 20422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-er 7949  df-en 8163  df-dom 8164  df-sdom 8165  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-nn 11277  df-2 11337  df-ndx 16136  df-slot 16137  df-base 16139  df-sets 16140  df-ress 16141  df-plusg 16230  df-0g 16371  df-mgm 17511  df-sgrp 17553  df-mnd 17564  df-grp 17695  df-minusg 17696  df-sbg 17697  df-subg 17858  df-ghm 17925  df-mgp 18760  df-ur 18772  df-ring 18819  df-lmod 19137  df-lss 19205  df-lsp 19247  df-lmhm 19297  df-lindf 20424
This theorem is referenced by:  lindsmm  20446
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