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Theorem lindslinindsimp2lem5 43919
Description: Lemma 5 for lindslinindsimp2 43920. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
lindslinind.r 𝑅 = (Scalar‘𝑀)
lindslinind.b 𝐵 = (Base‘𝑅)
lindslinind.0 0 = (0g𝑅)
lindslinind.z 𝑍 = (0g𝑀)
Assertion
Ref Expression
lindslinindsimp2lem5 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 )))
Distinct variable groups:   𝐵,𝑓,𝑔,𝑦   𝑓,𝑀,𝑔,𝑦   𝑅,𝑓,𝑥   𝑆,𝑓,𝑔,𝑥,𝑦   𝑔,𝑉,𝑦   𝑓,𝑍,𝑔,𝑦   0 ,𝑓,𝑔,𝑥,𝑦   𝑅,𝑔,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝑀(𝑥)   𝑉(𝑥,𝑓)   𝑍(𝑥)

Proof of Theorem lindslinindsimp2lem5
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax-1 6 . . 3 ((𝑓𝑥) = 0 → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 ))
212a1d 26 . 2 ((𝑓𝑥) = 0 → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 ))))
3 elmapi 8234 . . . . . . . . . 10 (𝑓 ∈ (𝐵𝑚 𝑆) → 𝑓:𝑆𝐵)
4 ffvelrn 6680 . . . . . . . . . . . . . 14 ((𝑓:𝑆𝐵𝑥𝑆) → (𝑓𝑥) ∈ 𝐵)
54expcom 406 . . . . . . . . . . . . 13 (𝑥𝑆 → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
65adantl 474 . . . . . . . . . . . 12 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
76adantl 474 . . . . . . . . . . 11 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
87com12 32 . . . . . . . . . 10 (𝑓:𝑆𝐵 → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
93, 8syl 17 . . . . . . . . 9 (𝑓 ∈ (𝐵𝑚 𝑆) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
109adantr 473 . . . . . . . 8 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
1110impcom 399 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓𝑥) ∈ 𝐵)
1211biantrurd 525 . . . . . 6 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → ((𝑓𝑥) ≠ 0 ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ≠ 0 )))
13 df-ne 2970 . . . . . . 7 ((𝑓𝑥) ≠ 0 ↔ ¬ (𝑓𝑥) = 0 )
1413bicomi 216 . . . . . 6 (¬ (𝑓𝑥) = 0 ↔ (𝑓𝑥) ≠ 0 )
15 eldifsn 4598 . . . . . 6 ((𝑓𝑥) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ≠ 0 ))
1612, 14, 153bitr4g 306 . . . . 5 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (¬ (𝑓𝑥) = 0 ↔ (𝑓𝑥) ∈ (𝐵 ∖ { 0 })))
17 lindslinind.r . . . . . . . . . . 11 𝑅 = (Scalar‘𝑀)
1817lmodfgrp 19377 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
1918adantl 474 . . . . . . . . 9 ((𝑆𝑉𝑀 ∈ LMod) → 𝑅 ∈ Grp)
2019adantr 473 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑅 ∈ Grp)
2120adantr 473 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑅 ∈ Grp)
22 lindslinind.b . . . . . . . 8 𝐵 = (Base‘𝑅)
23 lindslinind.0 . . . . . . . 8 0 = (0g𝑅)
24 eqid 2780 . . . . . . . 8 (invg𝑅) = (invg𝑅)
2522, 23, 24grpinvnzcl 17970 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝑓𝑥) ∈ (𝐵 ∖ { 0 })) → ((invg𝑅)‘(𝑓𝑥)) ∈ (𝐵 ∖ { 0 }))
2621, 25sylan 572 . . . . . 6 (((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) ∧ (𝑓𝑥) ∈ (𝐵 ∖ { 0 })) → ((invg𝑅)‘(𝑓𝑥)) ∈ (𝐵 ∖ { 0 }))
2726ex 405 . . . . 5 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → ((𝑓𝑥) ∈ (𝐵 ∖ { 0 }) → ((invg𝑅)‘(𝑓𝑥)) ∈ (𝐵 ∖ { 0 })))
2816, 27sylbid 232 . . . 4 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (¬ (𝑓𝑥) = 0 → ((invg𝑅)‘(𝑓𝑥)) ∈ (𝐵 ∖ { 0 })))
29 oveq1 6989 . . . . . . . . . . 11 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (𝑦( ·𝑠𝑀)𝑥) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
3029eqeq1d 2782 . . . . . . . . . 10 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → ((𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
3130notbid 310 . . . . . . . . 9 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
3231orbi2d 900 . . . . . . . 8 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → ((¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
3332ralbidv 3149 . . . . . . 7 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) ↔ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
3433rspcva 3535 . . . . . 6 ((((invg𝑅)‘(𝑓𝑥)) ∈ (𝐵 ∖ { 0 }) ∧ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))) → ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
35 simpl 475 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆𝑉𝑀 ∈ LMod))
3635adantr 473 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆𝑉𝑀 ∈ LMod))
37 simplrl 765 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑆 ⊆ (Base‘𝑀))
38 simplrr 766 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑥𝑆)
39 simpl 475 . . . . . . . . 9 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 ∈ (𝐵𝑚 𝑆))
4039adantl 474 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑓 ∈ (𝐵𝑚 𝑆))
41 lindslinind.z . . . . . . . . 9 𝑍 = (0g𝑀)
42 eqid 2780 . . . . . . . . 9 ((invg𝑅)‘(𝑓𝑥)) = ((invg𝑅)‘(𝑓𝑥))
43 eqid 2780 . . . . . . . . 9 (𝑓 ↾ (𝑆 ∖ {𝑥})) = (𝑓 ↾ (𝑆 ∖ {𝑥}))
4417, 22, 23, 41, 42, 43lindslinindimp2lem2 43916 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})))
4536, 37, 38, 40, 44syl13anc 1353 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})))
46 id 22 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → 𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})))
4723a1i 11 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → 0 = (0g𝑅))
4846, 47breq12d 4947 . . . . . . . . . . . . 13 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔 finSupp 0 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅)))
4948notbid 310 . . . . . . . . . . . 12 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ 𝑔 finSupp 0 ↔ ¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅)))
50 oveq1 6989 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))
5150eqeq2d 2790 . . . . . . . . . . . . 13 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → ((((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
5251notbid 310 . . . . . . . . . . . 12 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
5349, 52orbi12d 903 . . . . . . . . . . 11 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → ((¬ 𝑔 finSupp 0 ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) ↔ (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅) ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
5453rspcva 3535 . . . . . . . . . 10 (((𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})) ∧ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))) → (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅) ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
5523breq2i 4942 . . . . . . . . . . . . . . . . . 18 (𝑓 finSupp 0𝑓 finSupp (0g𝑅))
5655biimpi 208 . . . . . . . . . . . . . . . . 17 (𝑓 finSupp 0𝑓 finSupp (0g𝑅))
5756adantr 473 . . . . . . . . . . . . . . . 16 ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp (0g𝑅))
5857adantl 474 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 finSupp (0g𝑅))
5958adantl 474 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑓 finSupp (0g𝑅))
60 fvexd 6519 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (0g𝑅) ∈ V)
6159, 60fsuppres 8659 . . . . . . . . . . . . 13 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅))
6261pm2.24d 149 . . . . . . . . . . . 12 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅) → (𝑓𝑥) = 0 ))
6362com12 32 . . . . . . . . . . 11 (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅) → ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓𝑥) = 0 ))
64 simplr 757 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑀 ∈ LMod)
6564adantr 473 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑀 ∈ LMod)
6617fveq2i 6507 . . . . . . . . . . . . . . . . . 18 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
6722, 66eqtr2i 2805 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝑀)) = 𝐵
6867oveq1i 6992 . . . . . . . . . . . . . . . 16 ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑥})) = (𝐵𝑚 (𝑆 ∖ {𝑥}))
6945, 68syl6eleqr 2879 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑥})))
70 ssdifss 4004 . . . . . . . . . . . . . . . . . . 19 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
7170adantr 473 . . . . . . . . . . . . . . . . . 18 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
7271adantl 474 . . . . . . . . . . . . . . . . 17 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
73 difexg 5091 . . . . . . . . . . . . . . . . . . . 20 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
7473adantr 473 . . . . . . . . . . . . . . . . . . 19 ((𝑆𝑉𝑀 ∈ LMod) → (𝑆 ∖ {𝑥}) ∈ V)
7574adantr 473 . . . . . . . . . . . . . . . . . 18 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
76 elpwg 4433 . . . . . . . . . . . . . . . . . 18 ((𝑆 ∖ {𝑥}) ∈ V → ((𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
7775, 76syl 17 . . . . . . . . . . . . . . . . 17 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
7872, 77mpbird 249 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
7978adantr 473 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
80 lincval 43866 . . . . . . . . . . . . . . 15 ((𝑀 ∈ LMod ∧ (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑥})) ∧ (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀)) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})) = (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·𝑠𝑀)𝑧))))
8165, 69, 79, 80syl3anc 1352 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})) = (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·𝑠𝑀)𝑧))))
82 fvres 6523 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧) = (𝑓𝑧))
8382adantl 474 . . . . . . . . . . . . . . . . 17 (((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) ∧ 𝑧 ∈ (𝑆 ∖ {𝑥})) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧) = (𝑓𝑧))
8483oveq1d 6997 . . . . . . . . . . . . . . . 16 (((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) ∧ 𝑧 ∈ (𝑆 ∖ {𝑥})) → (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·𝑠𝑀)𝑧) = ((𝑓𝑧)( ·𝑠𝑀)𝑧))
8584mpteq2dva 5027 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·𝑠𝑀)𝑧)) = (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑧)( ·𝑠𝑀)𝑧)))
8685oveq2d 6998 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·𝑠𝑀)𝑧))) = (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑧)( ·𝑠𝑀)𝑧))))
87 simplr 757 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
88 3anass 1077 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
8988bicomi 216 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) ↔ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9089biimpi 208 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9190adantl 474 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9217, 22, 23, 41, 42, 43lindslinindimp2lem4 43918 . . . . . . . . . . . . . . 15 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑧)( ·𝑠𝑀)𝑧))) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
9336, 87, 91, 92syl3anc 1352 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑧)( ·𝑠𝑀)𝑧))) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
9481, 86, 933eqtrrd 2821 . . . . . . . . . . . . 13 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))
9594pm2.24d 149 . . . . . . . . . . . 12 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})) → (𝑓𝑥) = 0 ))
9695com12 32 . . . . . . . . . . 11 (¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})) → ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓𝑥) = 0 ))
9763, 96jaoi 844 . . . . . . . . . 10 ((¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅) ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓𝑥) = 0 ))
9854, 97syl 17 . . . . . . . . 9 (((𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})) ∧ ∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))) → ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓𝑥) = 0 ))
9998ex 405 . . . . . . . 8 ((𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})) → (∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓𝑥) = 0 )))
10099com23 86 . . . . . . 7 ((𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})) → ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 )))
10145, 100mpcom 38 . . . . . 6 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 ))
10234, 101syl5 34 . . . . 5 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → ((((invg𝑅)‘(𝑓𝑥)) ∈ (𝐵 ∖ { 0 }) ∧ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))) → (𝑓𝑥) = 0 ))
103102expd 408 . . . 4 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (((invg𝑅)‘(𝑓𝑥)) ∈ (𝐵 ∖ { 0 }) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 )))
10428, 103syldc 48 . . 3 (¬ (𝑓𝑥) = 0 → ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 )))
105104expd 408 . 2 (¬ (𝑓𝑥) = 0 → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 ))))
1062, 105pm2.61i 177 1 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wo 834  w3a 1069   = wceq 1508  wcel 2051  wne 2969  wral 3090  Vcvv 3417  cdif 3828  wss 3831  𝒫 cpw 4425  {csn 4444   class class class wbr 4934  cmpt 5013  cres 5413  wf 6189  cfv 6193  (class class class)co 6982  𝑚 cmap 8212   finSupp cfsupp 8634  Basecbs 16345  Scalarcsca 16430   ·𝑠 cvsca 16431  0gc0g 16575   Σg cgsu 16576  Grpcgrp 17903  invgcminusg 17904  LModclmod 19368   linC clinc 43861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-rep 5053  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285  ax-cnex 10397  ax-resscn 10398  ax-1cn 10399  ax-icn 10400  ax-addcl 10401  ax-addrcl 10402  ax-mulcl 10403  ax-mulrcl 10404  ax-mulcom 10405  ax-addass 10406  ax-mulass 10407  ax-distr 10408  ax-i2m1 10409  ax-1ne0 10410  ax-1rid 10411  ax-rnegex 10412  ax-rrecex 10413  ax-cnre 10414  ax-pre-lttri 10415  ax-pre-lttrn 10416  ax-pre-ltadd 10417  ax-pre-mulgt0 10418
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-tp 4449  df-op 4451  df-uni 4718  df-int 4755  df-iun 4799  df-iin 4800  df-br 4935  df-opab 4997  df-mpt 5014  df-tr 5036  df-id 5316  df-eprel 5321  df-po 5330  df-so 5331  df-fr 5370  df-se 5371  df-we 5372  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-pred 5991  df-ord 6037  df-on 6038  df-lim 6039  df-suc 6040  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-isom 6202  df-riota 6943  df-ov 6985  df-oprab 6986  df-mpo 6987  df-of 7233  df-om 7403  df-1st 7507  df-2nd 7508  df-supp 7640  df-wrecs 7756  df-recs 7818  df-rdg 7856  df-1o 7911  df-oadd 7915  df-er 8095  df-map 8214  df-en 8313  df-dom 8314  df-sdom 8315  df-fin 8316  df-fsupp 8635  df-oi 8775  df-card 9168  df-pnf 10482  df-mnf 10483  df-xr 10484  df-ltxr 10485  df-le 10486  df-sub 10678  df-neg 10679  df-nn 11446  df-2 11509  df-n0 11714  df-z 11800  df-uz 12065  df-fz 12715  df-fzo 12856  df-seq 13191  df-hash 13512  df-ndx 16348  df-slot 16349  df-base 16351  df-sets 16352  df-ress 16353  df-plusg 16440  df-0g 16577  df-gsum 16578  df-mre 16727  df-mrc 16728  df-acs 16730  df-mgm 17722  df-sgrp 17764  df-mnd 17775  df-submnd 17816  df-grp 17906  df-minusg 17907  df-mulg 18024  df-cntz 18230  df-cmn 18680  df-abl 18681  df-mgp 18975  df-ur 18987  df-ring 19034  df-lmod 19370  df-linc 43863
This theorem is referenced by:  lindslinindsimp2  43920
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