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Theorem lindslinindsimp2lem5 48307
Description: Lemma 5 for lindslinindsimp2 48308. (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 8887 . . . . . . . . . 10 (𝑓 ∈ (𝐵m 𝑆) → 𝑓:𝑆𝐵)
4 ffvelcdm 7100 . . . . . . . . . . . . . 14 ((𝑓:𝑆𝐵𝑥𝑆) → (𝑓𝑥) ∈ 𝐵)
54expcom 413 . . . . . . . . . . . . 13 (𝑥𝑆 → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
65adantl 481 . . . . . . . . . . . 12 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
76adantl 481 . . . . . . . . . . 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 480 . . . . . . . 8 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
1110impcom 407 . . . . . . 7 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓𝑥) ∈ 𝐵)
1211biantrurd 532 . . . . . 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 224 . . . . . 6 (¬ (𝑓𝑥) = 0 ↔ (𝑓𝑥) ≠ 0 )
15 eldifsn 4790 . . . . . 6 ((𝑓𝑥) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑓𝑥) ∈ 𝐵 ∧ (𝑓𝑥) ≠ 0 ))
1612, 14, 153bitr4g 314 . . . . 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 20883 . . . . . . . . . 10 (𝑀 ∈ LMod → 𝑅 ∈ Grp)
1918adantl 481 . . . . . . . . 9 ((𝑆𝑉𝑀 ∈ LMod) → 𝑅 ∈ Grp)
2019adantr 480 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑅 ∈ Grp)
2120adantr 480 . . . . . . 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 2734 . . . . . . . 8 (invg𝑅) = (invg𝑅)
2522, 23, 24grpinvnzcl 19041 . . . . . . 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 412 . . . . 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 240 . . . 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 7437 . . . . . . . . . . 11 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (𝑦( ·𝑠𝑀)𝑥) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
3029eqeq1d 2736 . . . . . . . . . 10 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → ((𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
3130notbid 318 . . . . . . . . 9 (𝑦 = ((invg𝑅)‘(𝑓𝑥)) → (¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ¬ (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
3231orbi2d 915 . . . . . . . 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 3175 . . . . . . 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 3619 . . . . . 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 482 . . . . . . . . 9 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆𝑉𝑀 ∈ LMod))
3635adantr 480 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆𝑉𝑀 ∈ LMod))
37 simplrl 777 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑆 ⊆ (Base‘𝑀))
38 simplrr 778 . . . . . . . 8 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑥𝑆)
39 simpl 482 . . . . . . . . 9 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 ∈ (𝐵m 𝑆))
4039adantl 481 . . . . . . . 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 2734 . . . . . . . . 9 ((invg𝑅)‘(𝑓𝑥)) = ((invg𝑅)‘(𝑓𝑥))
43 eqid 2734 . . . . . . . . 9 (𝑓 ↾ (𝑆 ∖ {𝑥})) = (𝑓 ↾ (𝑆 ∖ {𝑥}))
4417, 22, 23, 41, 42, 43lindslinindimp2lem2 48304 . . . . . . . 8 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵m 𝑆))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵m (𝑆 ∖ {𝑥})))
4536, 37, 38, 40, 44syl13anc 1371 . . . . . . 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 5160 . . . . . . . . . . . . 13 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔 finSupp 0 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅)))
4948notbid 318 . . . . . . . . . . . 12 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ 𝑔 finSupp 0 ↔ ¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g𝑅)))
50 oveq1 7437 . . . . . . . . . . . . . 14 (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))
5150eqeq2d 2745 . . . . . . . . . . . . 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 318 . . . . . . . . . . . 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 918 . . . . . . . . . . 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 3619 . . . . . . . . . 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 5155 . . . . . . . . . . . . . . . . . 18 (𝑓 finSupp 0𝑓 finSupp (0g𝑅))
5655biimpi 216 . . . . . . . . . . . . . . . . 17 (𝑓 finSupp 0𝑓 finSupp (0g𝑅))
5756adantr 480 . . . . . . . . . . . . . . . 16 ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp (0g𝑅))
5857adantl 481 . . . . . . . . . . . . . . 15 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 finSupp (0g𝑅))
5958adantl 481 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑓 finSupp (0g𝑅))
60 fvexd 6921 . . . . . . . . . . . . . 14 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (0g𝑅) ∈ V)
6159, 60fsuppres 9430 . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑀 ∈ LMod)
6564adantr 480 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑀 ∈ LMod)
6617fveq2i 6909 . . . . . . . . . . . . . . . . . 18 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
6722, 66eqtr2i 2763 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝑀)) = 𝐵
6867oveq1i 7440 . . . . . . . . . . . . . . . 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 4149 . . . . . . . . . . . . . . . . . . 19 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
7170adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
7271adantl 481 . . . . . . . . . . . . . . . . 17 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
73 difexg 5334 . . . . . . . . . . . . . . . . . . . 20 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
7473adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑆𝑉𝑀 ∈ LMod) → (𝑆 ∖ {𝑥}) ∈ V)
7574adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
76 elpwg 4607 . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
7978adantr 480 . . . . . . . . . . . . . . 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 48254 . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . 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 6925 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧) = (𝑓𝑧))
8382adantl 481 . . . . . . . . . . . . . . . . 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 7445 . . . . . . . . . . . . . . . 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 5247 . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . 15 ((((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
88 3anass 1094 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
8988bicomi 224 . . . . . . . . . . . . . . . . 17 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) ↔ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9089biimpi 216 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (𝐵m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
9190adantl 481 . . . . . . . . . . . . . . 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 48306 . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . 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 857 . . . . . . . . . 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 412 . . . . . . . 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 415 . . . 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 415 . 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 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  cdif 3959  wss 3962  𝒫 cpw 4604  {csn 4630   class class class wbr 5147  cmpt 5230  cres 5690  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864   finSupp cfsupp 9398  Basecbs 17244  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485   Σg cgsu 17486  Grpcgrp 18963  invgcminusg 18964  LModclmod 20874   linC clinc 48249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-grp 18966  df-minusg 18967  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-lmod 20876  df-linc 48251
This theorem is referenced by:  lindslinindsimp2  48308
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