Theorem lindslinindsimp2lem5 48587

Description: Lemma 5 for lindslinindsimp2 48588. (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 = (0g‘𝑅)
lindslinind.z	⊢ 𝑍 = (0g‘𝑀)

Assertion

Ref	Expression
lindslinindsimp2lem5	⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 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.

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ ((𝑓‘𝑥) = 0 → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 ))
2	1	2a1d 26	. 2 ⊢ ((𝑓‘𝑥) = 0 → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 ))))
3		elmapi 8779	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → 𝑓:𝑆⟶𝐵)
4		ffvelcdm 7020	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑆⟶𝐵 ∧ 𝑥 ∈ 𝑆) → (𝑓‘𝑥) ∈ 𝐵)
5	4	expcom 413	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑆 → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
6	5	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
7	6	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
8	7	com12 32	. . . . . . . . . 10 ⊢ (𝑓:𝑆⟶𝐵 → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓‘𝑥) ∈ 𝐵))
9	3, 8	syl 17	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓‘𝑥) ∈ 𝐵))
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓‘𝑥) ∈ 𝐵))
11	10	impcom 407	. . . . . . 7 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓‘𝑥) ∈ 𝐵)
12	11	biantrurd 532	. . . . . 6 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → ((𝑓‘𝑥) ≠ 0 ↔ ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ≠ 0 )))
13		df-ne 2930	. . . . . . 7 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) = 0 )
14	13	bicomi 224	. . . . . 6 ⊢ (¬ (𝑓‘𝑥) = 0 ↔ (𝑓‘𝑥) ≠ 0 )
15		eldifsn 4737	. . . . . 6 ⊢ ((𝑓‘𝑥) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ≠ 0 ))
16	12, 14, 15	3bitr4g 314	. . . . 5 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (¬ (𝑓‘𝑥) = 0 ↔ (𝑓‘𝑥) ∈ (𝐵 ∖ { 0 })))
17		lindslinind.r	. . . . . . . . . . 11 ⊢ 𝑅 = (Scalar‘𝑀)
18	17	lmodfgrp 20804	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
19	18	adantl 481	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑅 ∈ Grp)
20	19	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑅 ∈ Grp)
21	20	adantr 480	. . . . . . 7 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑅 ∈ Grp)
22		lindslinind.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
23		lindslinind.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑅)
24		eqid 2733	. . . . . . . 8 ⊢ (invg‘𝑅) = (invg‘𝑅)
25	22, 23, 24	grpinvnzcl 18926	. . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ (𝑓‘𝑥) ∈ (𝐵 ∖ { 0 })) → ((invg‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 }))
26	21, 25	sylan 580	. . . . . 6 ⊢ (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) ∧ (𝑓‘𝑥) ∈ (𝐵 ∖ { 0 })) → ((invg‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 }))
27	26	ex 412	. . . . 5 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → ((𝑓‘𝑥) ∈ (𝐵 ∖ { 0 }) → ((invg‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 })))
28	16, 27	sylbid 240	. . . 4 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (¬ (𝑓‘𝑥) = 0 → ((invg‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 })))
29		oveq1 7359	. . . . . . . . . . 11 ⊢ (𝑦 = ((invg‘𝑅)‘(𝑓‘𝑥)) → (𝑦( ·𝑠 ‘𝑀)𝑥) = (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥))
30	29	eqeq1d 2735	. . . . . . . . . 10 ⊢ (𝑦 = ((invg‘𝑅)‘(𝑓‘𝑥)) → ((𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
31	30	notbid 318	. . . . . . . . 9 ⊢ (𝑦 = ((invg‘𝑅)‘(𝑓‘𝑥)) → (¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
32	31	orbi2d 915	. . . . . . . 8 ⊢ (𝑦 = ((invg‘𝑅)‘(𝑓‘𝑥)) → ((¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
33	32	ralbidv 3156	. . . . . . 7 ⊢ (𝑦 = ((invg‘𝑅)‘(𝑓‘𝑥)) → (∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) ↔ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
34	33	rspcva 3571	. . . . . 6 ⊢ ((((invg‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 }) ∧ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))) → ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
35		simpl 482	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod))
36	35	adantr 480	. . . . . . . 8 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod))
37		simplrl 776	. . . . . . . 8 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑆 ⊆ (Base‘𝑀))
38		simplrr 777	. . . . . . . 8 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑥 ∈ 𝑆)
39		simpl 482	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 ∈ (𝐵 ↑m 𝑆))
40	39	adantl 481	. . . . . . . 8 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑓 ∈ (𝐵 ↑m 𝑆))
41		lindslinind.z	. . . . . . . . 9 ⊢ 𝑍 = (0g‘𝑀)
42		eqid 2733	. . . . . . . . 9 ⊢ ((invg‘𝑅)‘(𝑓‘𝑥)) = ((invg‘𝑅)‘(𝑓‘𝑥))
43		eqid 2733	. . . . . . . . 9 ⊢ (𝑓 ↾ (𝑆 ∖ {𝑥})) = (𝑓 ↾ (𝑆 ∖ {𝑥}))
44	17, 22, 23, 41, 42, 43	lindslinindimp2lem2 48584	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})))
45	36, 37, 38, 40, 44	syl13anc 1374	. . . . . . 7 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})))
46		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → 𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})))
47	23	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → 0 = (0g‘𝑅))
48	46, 47	breq12d 5106	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔 finSupp 0 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g‘𝑅)))
49	48	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ 𝑔 finSupp 0 ↔ ¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g‘𝑅)))
50		oveq1 7359	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))
51	50	eqeq2d 2744	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → ((((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
52	51	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
53	49, 52	orbi12d 918	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → ((¬ 𝑔 finSupp 0 ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) ↔ (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g‘𝑅) ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
54	53	rspcva 3571	. . . . . . . . . 10 ⊢ (((𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})) ∧ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))) → (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g‘𝑅) ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
55	23	breq2i 5101	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 finSupp 0 ↔ 𝑓 finSupp (0g‘𝑅))
56	55	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 finSupp 0 → 𝑓 finSupp (0g‘𝑅))
57	56	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp (0g‘𝑅))
58	57	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 finSupp (0g‘𝑅))
59	58	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑓 finSupp (0g‘𝑅))
60		fvexd 6843	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (0g‘𝑅) ∈ V)
61	59, 60	fsuppres 9284	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g‘𝑅))
62	61	pm2.24d 151	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g‘𝑅) → (𝑓‘𝑥) = 0 ))
63	62	com12 32	. . . . . . . . . . 11 ⊢ (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g‘𝑅) → ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓‘𝑥) = 0 ))
64		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑀 ∈ LMod)
65	64	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑀 ∈ LMod)
66	17	fveq2i 6831	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
67	22, 66	eqtr2i 2757	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑀)) = 𝐵
68	67	oveq1i 7362	. . . . . . . . . . . . . . . 16 ⊢ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥})) = (𝐵 ↑m (𝑆 ∖ {𝑥}))
69	45, 68	eleqtrrdi 2844	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥})))
70		ssdifss 4089	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
71	70	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
72	71	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
73		difexg 5269	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
74	73	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → (𝑆 ∖ {𝑥}) ∈ V)
75	74	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
76		elpwg 4552	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ∖ {𝑥}) ∈ V → ((𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
77	75, 76	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
78	72, 77	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
79	78	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
80		lincval 48534	. . . . . . . . . . . . . . 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)( ·𝑠 ‘𝑀)𝑧))))
81	65, 69, 79, 80	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 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)( ·𝑠 ‘𝑀)𝑧))))
82		fvres 6847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧) = (𝑓‘𝑧))
83	82	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) ∧ 𝑧 ∈ (𝑆 ∖ {𝑥})) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧) = (𝑓‘𝑧))
84	83	oveq1d 7367	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) ∧ 𝑧 ∈ (𝑆 ∖ {𝑥})) → (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·𝑠 ‘𝑀)𝑧) = ((𝑓‘𝑧)( ·𝑠 ‘𝑀)𝑧))
85	84	mpteq2dva 5186	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·𝑠 ‘𝑀)𝑧)) = (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑧)( ·𝑠 ‘𝑀)𝑧)))
86	85	oveq2d 7368	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·𝑠 ‘𝑀)𝑧))) = (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑧)( ·𝑠 ‘𝑀)𝑧))))
87		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆))
88		3anass 1094	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
89	88	bicomi 224	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) ↔ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
90	89	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
91	90	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
92	17, 22, 23, 41, 42, 43	lindslinindimp2lem4 48586	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑧)( ·𝑠 ‘𝑀)𝑧))) = (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥))
93	36, 87, 91, 92	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑧)( ·𝑠 ‘𝑀)𝑧))) = (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥))
94	81, 86, 93	3eqtrrd 2773	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))
95	94	pm2.24d 151	. . . . . . . . . . . 12 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})) → (𝑓‘𝑥) = 0 ))
96	95	com12 32	. . . . . . . . . . 11 ⊢ (¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})) → ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓‘𝑥) = 0 ))
97	63, 96	jaoi 857	. . . . . . . . . 10 ⊢ ((¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g‘𝑅) ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓‘𝑥) = 0 ))
98	54, 97	syl 17	. . . . . . . . 9 ⊢ (((𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})) ∧ ∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))) → ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓‘𝑥) = 0 ))
99	98	ex 412	. . . . . . . 8 ⊢ ((𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})) → (∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑓‘𝑥) = 0 )))
100	99	com23 86	. . . . . . 7 ⊢ ((𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})) → ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 )))
101	45, 100	mpcom 38	. . . . . 6 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 ))
102	34, 101	syl5 34	. . . . 5 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → ((((invg‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 }) ∧ ∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))) → (𝑓‘𝑥) = 0 ))
103	102	expd 415	. . . 4 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (((invg‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 }) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 )))
104	28, 103	syldc 48	. . 3 ⊢ (¬ (𝑓‘𝑥) = 0 → ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 )))
105	104	expd 415	. 2 ⊢ (¬ (𝑓‘𝑥) = 0 → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 ))))
106	2, 105	pm2.61i 182	1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑓 ∈ (𝐵 ↑m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  Vcvv 3437   ∖ cdif 3895   ⊆ wss 3898  𝒫 cpw 4549  {csn 4575   class class class wbr 5093   ↦ cmpt 5174   ↾ cres 5621  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   ↑_m cmap 8756   finSupp cfsupp 9252  Basecbs 17122  Scalarcsca 17166   ·_𝑠 cvsca 17167  0_gc0g 17345   Σ_g cgsu 17346  Grpcgrp 18848  inv_gcminusg 18849  LModclmod 20795   linC clinc 48529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-n0 12389  df-z 12476  df-uz 12739  df-fz 13410  df-fzo 13557  df-seq 13911  df-hash 14240  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-0g 17347  df-gsum 17348  df-mre 17490  df-mrc 17491  df-acs 17493  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-submnd 18694  df-grp 18851  df-minusg 18852  df-mulg 18983  df-cntz 19231  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-lmod 20797  df-linc 48531

This theorem is referenced by:  lindslinindsimp2  48588