Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lindslinindsimp2lem5 Structured version   Visualization version   GIF version

Theorem lindslinindsimp2lem5 44868
 Description: Lemma 5 for lindslinindsimp2 44869. (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 8415 . . . . . . . . . 10 (𝑓 ∈ (𝐵m 𝑆) → 𝑓:𝑆𝐵)
4 ffvelrn 6830 . . . . . . . . . . . . . 14 ((𝑓:𝑆𝐵𝑥𝑆) → (𝑓𝑥) ∈ 𝐵)
54expcom 417 . . . . . . . . . . . . 13 (𝑥𝑆 → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
65adantl 485 . . . . . . . . . . . 12 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
76adantl 485 . . . . . . . . . . 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 484 . . . . . . . 8 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
1110impcom 411 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓𝑥) ∈ 𝐵)
1211biantrurd 536 . . . . . 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 2991 . . . . . . 7 ((𝑓𝑥) ≠ 0 ↔ ¬ (𝑓𝑥) = 0 )
1413bicomi 227 . . . . . 6 (¬ (𝑓𝑥) = 0 ↔ (𝑓𝑥) ≠ 0 )
15 eldifsn 4683 . . . . . 6 ((𝑓𝑥) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ≠ 0 ))
1612, 14, 153bitr4g 317 . . . . 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 19640 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
1918adantl 485 . . . . . . . . 9 ((𝑆𝑉𝑀 ∈ LMod) → 𝑅 ∈ Grp)
2019adantr 484 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑅 ∈ Grp)
2120adantr 484 . . . . . . 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 2801 . . . . . . . 8 (invg𝑅) = (invg𝑅)
2522, 23, 24grpinvnzcl 18167 . . . . . . 7 ((𝑅 ∈ Grp ∧ (𝑓𝑥) ∈ (𝐵 ∖ { 0 })) → ((invg𝑅)‘(𝑓𝑥)) ∈ (𝐵 ∖ { 0 }))
2621, 25sylan 583 . . . . . 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 416 . . . . 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 243 . . . 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 7146 . . . . . . . . . . 11 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (𝑦( ·𝑠𝑀)𝑥) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
3029eqeq1d 2803 . . . . . . . . . 10 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → ((𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
3130notbid 321 . . . . . . . . 9 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
3231orbi2d 913 . . . . . . . 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 3165 . . . . . . 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 3572 . . . . . 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 486 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆𝑉𝑀 ∈ LMod))
3635adantr 484 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆𝑉𝑀 ∈ LMod))
37 simplrl 776 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑆 ⊆ (Base‘𝑀))
38 simplrr 777 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑥𝑆)
39 simpl 486 . . . . . . . . 9 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 ∈ (𝐵m 𝑆))
4039adantl 485 . . . . . . . 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 2801 . . . . . . . . 9 ((invg𝑅)‘(𝑓𝑥)) = ((invg𝑅)‘(𝑓𝑥))
43 eqid 2801 . . . . . . . . 9 (𝑓 ↾ (𝑆 ∖ {𝑥})) = (𝑓 ↾ (𝑆 ∖ {𝑥}))
4417, 22, 23, 41, 42, 43lindslinindimp2lem2 44865 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵m 𝑆))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵m (𝑆 ∖ {𝑥})))
4536, 37, 38, 40, 44syl13anc 1369 . . . . . . 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 5046 . . . . . . . . . . . . 13 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔 finSupp 0 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅)))
4948notbid 321 . . . . . . . . . . . 12 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ 𝑔 finSupp 0 ↔ ¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅)))
50 oveq1 7146 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))
5150eqeq2d 2812 . . . . . . . . . . . . 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 321 . . . . . . . . . . . 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 916 . . . . . . . . . . 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 3572 . . . . . . . . . 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 5041 . . . . . . . . . . . . . . . . . 18 (𝑓 finSupp 0𝑓 finSupp (0g𝑅))
5655biimpi 219 . . . . . . . . . . . . . . . . 17 (𝑓 finSupp 0𝑓 finSupp (0g𝑅))
5756adantr 484 . . . . . . . . . . . . . . . 16 ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp (0g𝑅))
5857adantl 485 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 finSupp (0g𝑅))
5958adantl 485 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑓 finSupp (0g𝑅))
60 fvexd 6664 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (0g𝑅) ∈ V)
6159, 60fsuppres 8846 . . . . . . . . . . . . 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 154 . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑀 ∈ LMod)
6564adantr 484 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑀 ∈ LMod)
6617fveq2i 6652 . . . . . . . . . . . . . . . . . 18 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
6722, 66eqtr2i 2825 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝑀)) = 𝐵
6867oveq1i 7149 . . . . . . . . . . . . . . . 16 ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥})) = (𝐵m (𝑆 ∖ {𝑥}))
6945, 68eleqtrrdi 2904 . . . . . . . . . . . . . . 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 4066 . . . . . . . . . . . . . . . . . . 19 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
7170adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
7271adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
73 difexg 5198 . . . . . . . . . . . . . . . . . . . 20 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
7473adantr 484 . . . . . . . . . . . . . . . . . . 19 ((𝑆𝑉𝑀 ∈ LMod) → (𝑆 ∖ {𝑥}) ∈ V)
7574adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
76 elpwg 4503 . . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
7978adantr 484 . . . . . . . . . . . . . . 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 44815 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 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 6668 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧) = (𝑓𝑧))
8382adantl 485 . . . . . . . . . . . . . . . . 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 7154 . . . . . . . . . . . . . . . 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 5128 . . . . . . . . . . . . . . 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 7155 . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
88 3anass 1092 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
8988bicomi 227 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) ↔ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9089biimpi 219 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9190adantl 485 . . . . . . . . . . . . . . 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 44867 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . 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 154 . . . . . . . . . . . 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 854 . . . . . . . . . 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 416 . . . . . . . 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 419 . . . 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 419 . 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 185 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  Vcvv 3444   ∖ cdif 3881   ⊆ wss 3884  𝒫 cpw 4500  {csn 4528   class class class wbr 5033   ↦ cmpt 5113   ↾ cres 5525  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393   finSupp cfsupp 8821  Basecbs 16479  Scalarcsca 16564   ·𝑠 cvsca 16565  0gc0g 16709   Σg cgsu 16710  Grpcgrp 18099  invgcminusg 18100  LModclmod 19631   linC clinc 44810 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-seq 13369  df-hash 13691  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-0g 16711  df-gsum 16712  df-mre 16853  df-mrc 16854  df-acs 16856  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-submnd 17953  df-grp 18102  df-minusg 18103  df-mulg 18221  df-cntz 18443  df-cmn 18904  df-abl 18905  df-mgp 19237  df-ur 19249  df-ring 19296  df-lmod 19633  df-linc 44812 This theorem is referenced by:  lindslinindsimp2  44869
 Copyright terms: Public domain W3C validator