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Theorem lindfmm 20653
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 20634 . . . . 5 Rel LIndF
21brrelex1i 5494 . . . 4 (𝐹 LIndF 𝑆𝐹 ∈ V)
3 simp3 1131 . . . 4 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → 𝐹:𝐼𝐵)
4 dmfex 7497 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝐼𝐵) → 𝐼 ∈ V)
52, 3, 4syl2anr 596 . . 3 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) ∧ 𝐹 LIndF 𝑆) → 𝐼 ∈ V)
65ex 413 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆𝐼 ∈ V))
71brrelex1i 5494 . . . 4 ((𝐺𝐹) LIndF 𝑇 → (𝐺𝐹) ∈ V)
8 f1f 6443 . . . . . 6 (𝐺:𝐵1-1𝐶𝐺:𝐵𝐶)
9 fco 6399 . . . . . 6 ((𝐺:𝐵𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
108, 9sylan 580 . . . . 5 ((𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
11103adant1 1123 . . . 4 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
12 dmfex 7497 . . . 4 (((𝐺𝐹) ∈ V ∧ (𝐺𝐹):𝐼𝐶) → 𝐼 ∈ V)
137, 11, 12syl2anr 596 . . 3 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) ∧ (𝐺𝐹) LIndF 𝑇) → 𝐼 ∈ V)
1413ex 413 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → ((𝐺𝐹) LIndF 𝑇𝐼 ∈ V))
15 eldifi 4024 . . . . . . . . 9 (𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
16 simpllr 772 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺:𝐵1-1𝐶)
17 lmhmlmod1 19495 . . . . . . . . . . . . . . 15 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
1817ad3antrrr 726 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑆 ∈ LMod)
19 simprr 769 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
20 simprl 767 . . . . . . . . . . . . . . 15 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐹:𝐼𝐵)
21 simpl 483 . . . . . . . . . . . . . . 15 ((𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆))) → 𝑥𝐼)
22 ffvelrn 6714 . . . . . . . . . . . . . . 15 ((𝐹:𝐼𝐵𝑥𝐼) → (𝐹𝑥) ∈ 𝐵)
2320, 21, 22syl2an 595 . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . 15 (Scalar‘𝑆) = (Scalar‘𝑆)
26 eqid 2795 . . . . . . . . . . . . . . 15 ( ·𝑠𝑆) = ( ·𝑠𝑆)
27 eqid 2795 . . . . . . . . . . . . . . 15 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
2824, 25, 26, 27lmodvscl 19341 . . . . . . . . . . . . . 14 ((𝑆 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑥) ∈ 𝐵) → (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵)
2918, 19, 23, 28syl3anc 1364 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵)
30 imassrn 5817 . . . . . . . . . . . . . . . 16 (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝐹
31 frn 6388 . . . . . . . . . . . . . . . . 17 (𝐹:𝐼𝐵 → ran 𝐹𝐵)
3231adantr 481 . . . . . . . . . . . . . . . 16 ((𝐹:𝐼𝐵𝐼 ∈ V) → ran 𝐹𝐵)
3330, 32syl5ss 3900 . . . . . . . . . . . . . . 15 ((𝐹:𝐼𝐵𝐼 ∈ V) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
3433ad2antlr 723 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
35 eqid 2795 . . . . . . . . . . . . . . 15 (LSpan‘𝑆) = (LSpan‘𝑆)
3624, 35lspssv 19445 . . . . . . . . . . . . . 14 ((𝑆 ∈ LMod ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
3718, 34, 36syl2anc 584 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
38 f1elima 6886 . . . . . . . . . . . . 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 1364 . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺 ∈ (𝑆 LMHom 𝑇))
41 eqid 2795 . . . . . . . . . . . . . . . 16 ( ·𝑠𝑇) = ( ·𝑠𝑇)
4225, 27, 24, 26, 41lmhmlin 19497 . . . . . . . . . . . . . . 15 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑥) ∈ 𝐵) → (𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) = (𝑘( ·𝑠𝑇)(𝐺‘(𝐹𝑥))))
4340, 19, 23, 42syl3anc 1364 . . . . . . . . . . . . . 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 6382 . . . . . . . . . . . . . . . . 17 (𝐹:𝐼𝐵𝐹 Fn 𝐼)
4544ad2antrl 724 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐹 Fn 𝐼)
46 fvco2 6625 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐼𝑥𝐼) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4745, 21, 46syl2an 595 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4847oveq2d 7032 . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . 16 (LSpan‘𝑇) = (LSpan‘𝑇)
5124, 35, 50lmhmlsp 19511 . . . . . . . . . . . . . . 15 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))))
5240, 34, 51syl2anc 584 . . . . . . . . . . . . . 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 5979 . . . . . . . . . . . . . . 15 ((𝐺𝐹) “ (𝐼 ∖ {𝑥})) = (𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))
5453fveq2i 6541 . . . . . . . . . . . . . 14 ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥}))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥}))))
5552, 54syl6eqr 2849 . . . . . . . . . . . . 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 2877 . . . . . . . . . . . 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 282 . . . . . . . . . . 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 319 . . . . . . . . . 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 468 . . . . . . . . 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 592 . . . . . . . 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 3163 . . . . . . 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 2795 . . . . . . . . . . . 12 (Scalar‘𝑇) = (Scalar‘𝑇)
6325, 62lmhmsca 19492 . . . . . . . . . . 11 (𝐺 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
6463fveq2d 6542 . . . . . . . . . 10 (𝐺 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
6563fveq2d 6542 . . . . . . . . . . 11 (𝐺 ∈ (𝑆 LMHom 𝑇) → (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑆)))
6665sneqd 4484 . . . . . . . . . 10 (𝐺 ∈ (𝑆 LMHom 𝑇) → {(0g‘(Scalar‘𝑇))} = {(0g‘(Scalar‘𝑆))})
6764, 66difeq12d 4021 . . . . . . . . 9 (𝐺 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}))
6867ad3antrrr 726 . . . . . . . 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 3375 . . . . . . 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 283 . . . . . 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 3163 . . . . 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 722 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝑆 ∈ LMod)
73 simprr 769 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐼 ∈ V)
74 eqid 2795 . . . . . . 7 (0g‘(Scalar‘𝑆)) = (0g‘(Scalar‘𝑆))
7524, 26, 35, 25, 27, 74islindf2 20640 . . . . . 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 1364 . . . . 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 19494 . . . . . . 7 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7877ad2antrr 722 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝑇 ∈ LMod)
7910ad2ant2lr 744 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐺𝐹):𝐼𝐶)
80 lindfmm.c . . . . . . 7 𝐶 = (Base‘𝑇)
81 eqid 2795 . . . . . . 7 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
82 eqid 2795 . . . . . . 7 (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑇))
8380, 41, 50, 62, 81, 82islindf2 20640 . . . . . 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 1364 . . . . 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 312 . . . 4 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))
8685exp32 421 . . 3 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) → (𝐹:𝐼𝐵 → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))))
87863impia 1110 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇)))
886, 14, 87pm5.21ndd 381 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  Vcvv 3437  cdif 3856  wss 3859  {csn 4472   class class class wbr 4962  ran crn 5444  cima 5446  ccom 5447   Fn wfn 6220  wf 6221  1-1wf1 6222  cfv 6225  (class class class)co 7016  Basecbs 16312  Scalarcsca 16397   ·𝑠 cvsca 16398  0gc0g 16542  LModclmod 19324  LSpanclspn 19433   LMHom clmhm 19481   LIndF clindf 20630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-0g 16544  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-grp 17864  df-minusg 17865  df-sbg 17866  df-subg 18030  df-ghm 18097  df-mgp 18930  df-ur 18942  df-ring 18989  df-lmod 19326  df-lss 19394  df-lsp 19434  df-lmhm 19484  df-lindf 20632
This theorem is referenced by:  lindsmm  20654
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