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Theorem lindfmm 21827
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 21808 . . . . 5 Rel LIndF
21brrelex1i 5740 . . . 4 (𝐹 LIndF 𝑆𝐹 ∈ V)
3 simp3 1135 . . . 4 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → 𝐹:𝐼𝐵)
4 dmfex 7920 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝐼𝐵) → 𝐼 ∈ V)
52, 3, 4syl2anr 595 . . 3 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) ∧ 𝐹 LIndF 𝑆) → 𝐼 ∈ V)
65ex 411 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆𝐼 ∈ V))
71brrelex1i 5740 . . . 4 ((𝐺𝐹) LIndF 𝑇 → (𝐺𝐹) ∈ V)
8 f1f 6800 . . . . . 6 (𝐺:𝐵1-1𝐶𝐺:𝐵𝐶)
9 fco 6754 . . . . . 6 ((𝐺:𝐵𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
108, 9sylan 578 . . . . 5 ((𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
11103adant1 1127 . . . 4 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
12 dmfex 7920 . . . 4 (((𝐺𝐹) ∈ V ∧ (𝐺𝐹):𝐼𝐶) → 𝐼 ∈ V)
137, 11, 12syl2anr 595 . . 3 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) ∧ (𝐺𝐹) LIndF 𝑇) → 𝐼 ∈ V)
1413ex 411 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → ((𝐺𝐹) LIndF 𝑇𝐼 ∈ V))
15 eldifi 4126 . . . . . . . . 9 (𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
16 simpllr 774 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺:𝐵1-1𝐶)
17 lmhmlmod1 21013 . . . . . . . . . . . . . . 15 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
1817ad3antrrr 728 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑆 ∈ LMod)
19 simprr 771 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
20 simprl 769 . . . . . . . . . . . . . . 15 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐹:𝐼𝐵)
21 simpl 481 . . . . . . . . . . . . . . 15 ((𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆))) → 𝑥𝐼)
22 ffvelcdm 7097 . . . . . . . . . . . . . . 15 ((𝐹:𝐼𝐵𝑥𝐼) → (𝐹𝑥) ∈ 𝐵)
2320, 21, 22syl2an 594 . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . 15 (Scalar‘𝑆) = (Scalar‘𝑆)
26 eqid 2726 . . . . . . . . . . . . . . 15 ( ·𝑠𝑆) = ( ·𝑠𝑆)
27 eqid 2726 . . . . . . . . . . . . . . 15 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
2824, 25, 26, 27lmodvscl 20856 . . . . . . . . . . . . . 14 ((𝑆 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑥) ∈ 𝐵) → (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵)
2918, 19, 23, 28syl3anc 1368 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵)
30 imassrn 6082 . . . . . . . . . . . . . . . 16 (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝐹
31 frn 6737 . . . . . . . . . . . . . . . . 17 (𝐹:𝐼𝐵 → ran 𝐹𝐵)
3231adantr 479 . . . . . . . . . . . . . . . 16 ((𝐹:𝐼𝐵𝐼 ∈ V) → ran 𝐹𝐵)
3330, 32sstrid 3991 . . . . . . . . . . . . . . 15 ((𝐹:𝐼𝐵𝐼 ∈ V) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
3433ad2antlr 725 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
35 eqid 2726 . . . . . . . . . . . . . . 15 (LSpan‘𝑆) = (LSpan‘𝑆)
3624, 35lspssv 20962 . . . . . . . . . . . . . 14 ((𝑆 ∈ LMod ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
3718, 34, 36syl2anc 582 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
38 f1elima 7280 . . . . . . . . . . . . 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 1368 . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺 ∈ (𝑆 LMHom 𝑇))
41 eqid 2726 . . . . . . . . . . . . . . . 16 ( ·𝑠𝑇) = ( ·𝑠𝑇)
4225, 27, 24, 26, 41lmhmlin 21015 . . . . . . . . . . . . . . 15 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑥) ∈ 𝐵) → (𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) = (𝑘( ·𝑠𝑇)(𝐺‘(𝐹𝑥))))
4340, 19, 23, 42syl3anc 1368 . . . . . . . . . . . . . 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 6730 . . . . . . . . . . . . . . . . 17 (𝐹:𝐼𝐵𝐹 Fn 𝐼)
4544ad2antrl 726 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐹 Fn 𝐼)
46 fvco2 7001 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐼𝑥𝐼) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4745, 21, 46syl2an 594 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4847oveq2d 7442 . . . . . . . . . . . . . 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 2769 . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . 16 (LSpan‘𝑇) = (LSpan‘𝑇)
5124, 35, 50lmhmlsp 21029 . . . . . . . . . . . . . . 15 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))))
5240, 34, 51syl2anc 582 . . . . . . . . . . . . . 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 6264 . . . . . . . . . . . . . . 15 ((𝐺𝐹) “ (𝐼 ∖ {𝑥})) = (𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))
5453fveq2i 6906 . . . . . . . . . . . . . 14 ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥}))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥}))))
5552, 54eqtr4di 2784 . . . . . . . . . . . . 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 2820 . . . . . . . . . . . 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 280 . . . . . . . . . . 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 317 . . . . . . . . . 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 466 . . . . . . . . 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 591 . . . . . . . 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 3166 . . . . . . 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 2726 . . . . . . . . . . . 12 (Scalar‘𝑇) = (Scalar‘𝑇)
6325, 62lmhmsca 21010 . . . . . . . . . . 11 (𝐺 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
6463fveq2d 6907 . . . . . . . . . 10 (𝐺 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
6563fveq2d 6907 . . . . . . . . . . 11 (𝐺 ∈ (𝑆 LMHom 𝑇) → (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑆)))
6665sneqd 4645 . . . . . . . . . 10 (𝐺 ∈ (𝑆 LMHom 𝑇) → {(0g‘(Scalar‘𝑇))} = {(0g‘(Scalar‘𝑆))})
6764, 66difeq12d 4122 . . . . . . . . 9 (𝐺 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}))
6867ad3antrrr 728 . . . . . . . 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 3315 . . . . . . 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 281 . . . . . 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 3166 . . . . 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 724 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝑆 ∈ LMod)
73 simprr 771 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐼 ∈ V)
74 eqid 2726 . . . . . . 7 (0g‘(Scalar‘𝑆)) = (0g‘(Scalar‘𝑆))
7524, 26, 35, 25, 27, 74islindf2 21814 . . . . . 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 1368 . . . . 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 21012 . . . . . . 7 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7877ad2antrr 724 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝑇 ∈ LMod)
7910ad2ant2lr 746 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐺𝐹):𝐼𝐶)
80 lindfmm.c . . . . . . 7 𝐶 = (Base‘𝑇)
81 eqid 2726 . . . . . . 7 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
82 eqid 2726 . . . . . . 7 (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑇))
8380, 41, 50, 62, 81, 82islindf2 21814 . . . . . 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 1368 . . . . 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 310 . . . 4 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))
8685exp32 419 . . 3 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) → (𝐹:𝐼𝐵 → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))))
87863impia 1114 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇)))
886, 14, 87pm5.21ndd 378 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 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  Vcvv 3462  cdif 3944  wss 3947  {csn 4633   class class class wbr 5155  ran crn 5685  cima 5687  ccom 5688   Fn wfn 6551  wf 6552  1-1wf1 6553  cfv 6556  (class class class)co 7426  Basecbs 17215  Scalarcsca 17271   ·𝑠 cvsca 17272  0gc0g 17456  LModclmod 20838  LSpanclspn 20950   LMHom clmhm 20999   LIndF clindf 21804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5292  ax-sep 5306  ax-nul 5313  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-cnex 11216  ax-resscn 11217  ax-1cn 11218  ax-icn 11219  ax-addcl 11220  ax-addrcl 11221  ax-mulcl 11222  ax-mulrcl 11223  ax-mulcom 11224  ax-addass 11225  ax-mulass 11226  ax-distr 11227  ax-i2m1 11228  ax-1ne0 11229  ax-1rid 11230  ax-rnegex 11231  ax-rrecex 11232  ax-cnre 11233  ax-pre-lttri 11234  ax-pre-lttrn 11235  ax-pre-ltadd 11236  ax-pre-mulgt0 11237
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-int 4957  df-iun 5005  df-br 5156  df-opab 5218  df-mpt 5239  df-tr 5273  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6314  df-ord 6381  df-on 6382  df-lim 6383  df-suc 6384  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8005  df-2nd 8006  df-frecs 8298  df-wrecs 8329  df-recs 8403  df-rdg 8442  df-er 8736  df-map 8859  df-en 8977  df-dom 8978  df-sdom 8979  df-pnf 11302  df-mnf 11303  df-xr 11304  df-ltxr 11305  df-le 11306  df-sub 11498  df-neg 11499  df-nn 12267  df-2 12329  df-sets 17168  df-slot 17186  df-ndx 17198  df-base 17216  df-ress 17245  df-plusg 17281  df-0g 17458  df-mgm 18635  df-sgrp 18714  df-mnd 18730  df-grp 18933  df-minusg 18934  df-sbg 18935  df-subg 19119  df-ghm 19209  df-mgp 20120  df-ur 20167  df-ring 20220  df-lmod 20840  df-lss 20911  df-lsp 20951  df-lmhm 21002  df-lindf 21806
This theorem is referenced by:  lindsmm  21828
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