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Theorem lindslinindsimp2lem5 46533
Description: Lemma 5 for lindslinindsimp2 46534. (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 8787 . . . . . . . . . 10 (𝑓 ∈ (𝐵m 𝑆) → 𝑓:𝑆𝐵)
4 ffvelcdm 7032 . . . . . . . . . . . . . 14 ((𝑓:𝑆𝐵𝑥𝑆) → (𝑓𝑥) ∈ 𝐵)
54expcom 414 . . . . . . . . . . . . 13 (𝑥𝑆 → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
65adantl 482 . . . . . . . . . . . 12 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
76adantl 482 . . . . . . . . . . 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 481 . . . . . . . 8 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
1110impcom 408 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓𝑥) ∈ 𝐵)
1211biantrurd 533 . . . . . 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 2944 . . . . . . 7 ((𝑓𝑥) ≠ 0 ↔ ¬ (𝑓𝑥) = 0 )
1413bicomi 223 . . . . . 6 (¬ (𝑓𝑥) = 0 ↔ (𝑓𝑥) ≠ 0 )
15 eldifsn 4747 . . . . . 6 ((𝑓𝑥) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ≠ 0 ))
1612, 14, 153bitr4g 313 . . . . 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 20331 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
1918adantl 482 . . . . . . . . 9 ((𝑆𝑉𝑀 ∈ LMod) → 𝑅 ∈ Grp)
2019adantr 481 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑅 ∈ Grp)
2120adantr 481 . . . . . . 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 2736 . . . . . . . 8 (invg𝑅) = (invg𝑅)
2522, 23, 24grpinvnzcl 18819 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝑓𝑥) ∈ (𝐵 ∖ { 0 })) → ((invg𝑅)‘(𝑓𝑥)) ∈ (𝐵 ∖ { 0 }))
2621, 25sylan 580 . . . . . 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 413 . . . . 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 239 . . . 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 7364 . . . . . . . . . . 11 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (𝑦( ·𝑠𝑀)𝑥) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
3029eqeq1d 2738 . . . . . . . . . 10 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → ((𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
3130notbid 317 . . . . . . . . 9 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
3231orbi2d 914 . . . . . . . 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 3174 . . . . . . 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 3579 . . . . . 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 483 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆𝑉𝑀 ∈ LMod))
3635adantr 481 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆𝑉𝑀 ∈ LMod))
37 simplrl 775 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑆 ⊆ (Base‘𝑀))
38 simplrr 776 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑥𝑆)
39 simpl 483 . . . . . . . . 9 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 ∈ (𝐵m 𝑆))
4039adantl 482 . . . . . . . 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 2736 . . . . . . . . 9 ((invg𝑅)‘(𝑓𝑥)) = ((invg𝑅)‘(𝑓𝑥))
43 eqid 2736 . . . . . . . . 9 (𝑓 ↾ (𝑆 ∖ {𝑥})) = (𝑓 ↾ (𝑆 ∖ {𝑥}))
4417, 22, 23, 41, 42, 43lindslinindimp2lem2 46530 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵m 𝑆))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵m (𝑆 ∖ {𝑥})))
4536, 37, 38, 40, 44syl13anc 1372 . . . . . . 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 5118 . . . . . . . . . . . . 13 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔 finSupp 0 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅)))
4948notbid 317 . . . . . . . . . . . 12 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ 𝑔 finSupp 0 ↔ ¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅)))
50 oveq1 7364 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))
5150eqeq2d 2747 . . . . . . . . . . . . 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 317 . . . . . . . . . . . 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 917 . . . . . . . . . . 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 3579 . . . . . . . . . 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 5113 . . . . . . . . . . . . . . . . . 18 (𝑓 finSupp 0𝑓 finSupp (0g𝑅))
5655biimpi 215 . . . . . . . . . . . . . . . . 17 (𝑓 finSupp 0𝑓 finSupp (0g𝑅))
5756adantr 481 . . . . . . . . . . . . . . . 16 ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp (0g𝑅))
5857adantl 482 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 finSupp (0g𝑅))
5958adantl 482 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑓 finSupp (0g𝑅))
60 fvexd 6857 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (0g𝑅) ∈ V)
6159, 60fsuppres 9330 . . . . . . . . . . . . 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 151 . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑀 ∈ LMod)
6564adantr 481 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑀 ∈ LMod)
6617fveq2i 6845 . . . . . . . . . . . . . . . . . 18 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
6722, 66eqtr2i 2765 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝑀)) = 𝐵
6867oveq1i 7367 . . . . . . . . . . . . . . . 16 ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥})) = (𝐵m (𝑆 ∖ {𝑥}))
6945, 68eleqtrrdi 2849 . . . . . . . . . . . . . . 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 4095 . . . . . . . . . . . . . . . . . . 19 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
7170adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
7271adantl 482 . . . . . . . . . . . . . . . . 17 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
73 difexg 5284 . . . . . . . . . . . . . . . . . . . 20 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
7473adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑆𝑉𝑀 ∈ LMod) → (𝑆 ∖ {𝑥}) ∈ V)
7574adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
76 elpwg 4563 . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
7978adantr 481 . . . . . . . . . . . . . . 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 46480 . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . 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 6861 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧) = (𝑓𝑧))
8382adantl 482 . . . . . . . . . . . . . . . . 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 7372 . . . . . . . . . . . . . . . 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 5205 . . . . . . . . . . . . . . 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 7373 . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
88 3anass 1095 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
8988bicomi 223 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) ↔ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9089biimpi 215 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9190adantl 482 . . . . . . . . . . . . . . 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 46532 . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . 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 151 . . . . . . . . . . . 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 855 . . . . . . . . . 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 413 . . . . . . . 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 416 . . . 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 416 . 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 182 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  Vcvv 3445  cdif 3907  wss 3910  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cmpt 5188  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765   finSupp cfsupp 9305  Basecbs 17083  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321   Σg cgsu 17322  Grpcgrp 18748  invgcminusg 18749  LModclmod 20322   linC clinc 46475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-lmod 20324  df-linc 46477
This theorem is referenced by:  lindslinindsimp2  46534
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