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Theorem lindslinindsimp2lem5 42920
Description: Lemma 5 for lindslinindsimp2 42921. (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 8082 . . . . . . . . . 10 (𝑓 ∈ (𝐵𝑚 𝑆) → 𝑓:𝑆𝐵)
4 ffvelrn 6547 . . . . . . . . . . . . . 14 ((𝑓:𝑆𝐵𝑥𝑆) → (𝑓𝑥) ∈ 𝐵)
54expcom 402 . . . . . . . . . . . . 13 (𝑥𝑆 → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
65adantl 473 . . . . . . . . . . . 12 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
76adantl 473 . . . . . . . . . . 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 472 . . . . . . . 8 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
1110impcom 396 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓𝑥) ∈ 𝐵)
1211biantrurd 528 . . . . . 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 2938 . . . . . . 7 ((𝑓𝑥) ≠ 0 ↔ ¬ (𝑓𝑥) = 0 )
1413bicomi 215 . . . . . 6 (¬ (𝑓𝑥) = 0 ↔ (𝑓𝑥) ≠ 0 )
15 eldifsn 4472 . . . . . 6 ((𝑓𝑥) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ≠ 0 ))
1612, 14, 153bitr4g 305 . . . . 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 19141 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
1918adantl 473 . . . . . . . . 9 ((𝑆𝑉𝑀 ∈ LMod) → 𝑅 ∈ Grp)
2019adantr 472 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑅 ∈ Grp)
2120adantr 472 . . . . . . 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 2765 . . . . . . . 8 (invg𝑅) = (invg𝑅)
2522, 23, 24grpinvnzcl 17754 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝑓𝑥) ∈ (𝐵 ∖ { 0 })) → ((invg𝑅)‘(𝑓𝑥)) ∈ (𝐵 ∖ { 0 }))
2621, 25sylan 575 . . . . . 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 401 . . . . 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 231 . . . 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 6849 . . . . . . . . . . 11 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (𝑦( ·𝑠𝑀)𝑥) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
3029eqeq1d 2767 . . . . . . . . . 10 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → ((𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
3130notbid 309 . . . . . . . . 9 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
3231orbi2d 939 . . . . . . . 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 3133 . . . . . . 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 3459 . . . . . 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 474 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆𝑉𝑀 ∈ LMod))
3635adantr 472 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆𝑉𝑀 ∈ LMod))
37 simplrl 795 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑆 ⊆ (Base‘𝑀))
38 simplrr 796 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑥𝑆)
39 simpl 474 . . . . . . . . 9 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 ∈ (𝐵𝑚 𝑆))
4039adantl 473 . . . . . . . 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 2765 . . . . . . . . 9 ((invg𝑅)‘(𝑓𝑥)) = ((invg𝑅)‘(𝑓𝑥))
43 eqid 2765 . . . . . . . . 9 (𝑓 ↾ (𝑆 ∖ {𝑥})) = (𝑓 ↾ (𝑆 ∖ {𝑥}))
4417, 22, 23, 41, 42, 43lindslinindimp2lem2 42917 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})))
4536, 37, 38, 40, 44syl13anc 1491 . . . . . . 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 4822 . . . . . . . . . . . . 13 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔 finSupp 0 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅)))
4948notbid 309 . . . . . . . . . . . 12 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ 𝑔 finSupp 0 ↔ ¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅)))
50 oveq1 6849 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))
5150eqeq2d 2775 . . . . . . . . . . . . 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 309 . . . . . . . . . . . 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 942 . . . . . . . . . . 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 3459 . . . . . . . . . 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 4817 . . . . . . . . . . . . . . . . . 18 (𝑓 finSupp 0𝑓 finSupp (0g𝑅))
5655biimpi 207 . . . . . . . . . . . . . . . . 17 (𝑓 finSupp 0𝑓 finSupp (0g𝑅))
5756adantr 472 . . . . . . . . . . . . . . . 16 ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp (0g𝑅))
5857adantl 473 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 finSupp (0g𝑅))
5958adantl 473 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑓 finSupp (0g𝑅))
60 fvexd 6390 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (0g𝑅) ∈ V)
6159, 60fsuppres 8507 . . . . . . . . . . . . 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 148 . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑀 ∈ LMod)
6564adantr 472 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑀 ∈ LMod)
6617fveq2i 6378 . . . . . . . . . . . . . . . . . 18 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
6722, 66eqtr2i 2788 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝑀)) = 𝐵
6867oveq1i 6852 . . . . . . . . . . . . . . . 16 ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑥})) = (𝐵𝑚 (𝑆 ∖ {𝑥}))
6945, 68syl6eleqr 2855 . . . . . . . . . . . . . . 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 3903 . . . . . . . . . . . . . . . . . . 19 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
7170adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
7271adantl 473 . . . . . . . . . . . . . . . . 17 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
73 difexg 4969 . . . . . . . . . . . . . . . . . . . 20 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
7473adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝑆𝑉𝑀 ∈ LMod) → (𝑆 ∖ {𝑥}) ∈ V)
7574adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
76 elpwg 4323 . . . . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
7978adantr 472 . . . . . . . . . . . . . . 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 42867 . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . 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 6394 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧) = (𝑓𝑧))
8382adantl 473 . . . . . . . . . . . . . . . . 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 6857 . . . . . . . . . . . . . . . 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 4903 . . . . . . . . . . . . . . 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 6858 . . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
88 3anass 1116 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
8988bicomi 215 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) ↔ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9089biimpi 207 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9190adantl 473 . . . . . . . . . . . . . . 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 42919 . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . 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 2804 . . . . . . . . . . . . 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 148 . . . . . . . . . . . 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 883 . . . . . . . . . 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 401 . . . . . . . 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 404 . . . 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 404 . 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 176 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  Vcvv 3350  cdif 3729  wss 3732  𝒫 cpw 4315  {csn 4334   class class class wbr 4809  cmpt 4888  cres 5279  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060   finSupp cfsupp 8482  Basecbs 16130  Scalarcsca 16217   ·𝑠 cvsca 16218  0gc0g 16366   Σg cgsu 16367  Grpcgrp 17689  invgcminusg 17690  LModclmod 19132   linC clinc 42862
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 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
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 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-0g 16368  df-gsum 16369  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-submnd 17602  df-grp 17692  df-minusg 17693  df-mulg 17808  df-cntz 18013  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-lmod 19134  df-linc 42864
This theorem is referenced by:  lindslinindsimp2  42921
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