Theorem lindslinindsimp2lem5 42045

Description: Lemma 5 for lindslinindsimp2 42046. (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 7765	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝑆) → 𝑓:𝑆⟶𝐵)
4		ffvelrn 6265	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝑆⟶𝐵 ∧ 𝑥 ∈ 𝑆) → (𝑓‘𝑥) ∈ 𝐵)
5	4	expcom 450	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑆 → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
6	5	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
7	6	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
8	7	com12 32	. . . . . . . . . . 11 ⊢ (𝑓:𝑆⟶𝐵 → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓‘𝑥) ∈ 𝐵))
9	3, 8	syl 17	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝑆) → (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑓‘𝑥) ∈ 𝐵))
10	9	adantr 480	. . . . . . . . 9 ⊢ ((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 445	. . . . . . . 8 ⊢ ((((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 528	. . . . . . 7 ⊢ ((((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 2782	. . . . . . . 8 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) = 0 )
14	13	bicomi 213	. . . . . . 7 ⊢ (¬ (𝑓‘𝑥) = 0 ↔ (𝑓‘𝑥) ≠ 0 )
15		eldifsn 4260	. . . . . . 7 ⊢ ((𝑓‘𝑥) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ≠ 0 ))
16	12, 14, 15	3bitr4g 302	. . . . . 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 ∖ { 0 })))
17		lindslinind.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Scalar‘𝑀)
18	17	lmodfgrp 18695	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
19	18	adantl 481	. . . . . . . . . 10 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑅 ∈ Grp)
20	19	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑅 ∈ Grp)
21	20	adantr 480	. . . . . . . 8 ⊢ ((((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	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
23		lindslinind.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
24		eqid 2610	. . . . . . . . 9 ⊢ (invg‘𝑅) = (invg‘𝑅)
25	22, 23, 24	grpinvnzcl 17310	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ (𝑓‘𝑥) ∈ (𝐵 ∖ { 0 })) → ((invg‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 }))
26	21, 25	sylan 487	. . . . . . 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 ∖ { 0 })) → ((invg‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 }))
27	26	ex 449	. . . . . 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 })))
28	16, 27	sylbid 229	. . . . 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 → ((invg‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 })))
29		oveq1 6556	. . . . . . . . . . . 12 ⊢ (𝑦 = ((invg‘𝑅)‘(𝑓‘𝑥)) → (𝑦( ·𝑠 ‘𝑀)𝑥) = (((invg‘𝑅)‘(𝑓‘𝑥))( ·𝑠 ‘𝑀)𝑥))
30	29	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (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 307	. . . . . . . . . 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}))))
32	31	orbi2d 734	. . . . . . . . 9 ⊢ (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 2969	. . . . . . . 8 ⊢ (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 3280	. . . . . . 7 ⊢ ((((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 472	. . . . . . . . . 10 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod))
36	35	adantr 480	. . . . . . . . 9 ⊢ ((((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 796	. . . . . . . . 9 ⊢ ((((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 797	. . . . . . . . 9 ⊢ ((((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 472	. . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐵 ↑𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 ∈ (𝐵 ↑𝑚 𝑆))
40	39	adantl 481	. . . . . . . . 9 ⊢ ((((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	. . . . . . . . . 10 ⊢ 𝑍 = (0g‘𝑀)
42		eqid 2610	. . . . . . . . . 10 ⊢ ((invg‘𝑅)‘(𝑓‘𝑥)) = ((invg‘𝑅)‘(𝑓‘𝑥))
43		eqid 2610	. . . . . . . . . 10 ⊢ (𝑓 ↾ (𝑆 ∖ {𝑥})) = (𝑓 ↾ (𝑆 ∖ {𝑥}))
44	17, 22, 23, 41, 42, 43	lindslinindimp2lem2 42042	. . . . . . . . 9 ⊢ (((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 1320	. . . . . . . 8 ⊢ ((((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	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → 𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})))
47	23	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → 0 = (0g‘𝑅))
48	46, 47	breq12d 4596	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔 finSupp 0 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g‘𝑅)))
49	48	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ 𝑔 finSupp 0 ↔ ¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0g‘𝑅)))
50		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))
51	50	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (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 307	. . . . . . . . . . . . 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}))))
53	49, 52	orbi12d 742	. . . . . . . . . . . 12 ⊢ (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 3280	. . . . . . . . . . 11 ⊢ (((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 4591	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 finSupp 0 ↔ 𝑓 finSupp (0g‘𝑅))
56	55	biimpi 205	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 finSupp 0 → 𝑓 finSupp (0g‘𝑅))
57	56	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp (0g‘𝑅))
58	57	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ (𝐵 ↑𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 finSupp (0g‘𝑅))
59	58	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 finSupp (0g‘𝑅))
60		fvex 6113	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) ∈ V
61	60	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (0g‘𝑅) ∈ V)
62	59, 61	fsuppres 8183	. . . . . . . . . . . . . 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})) finSupp (0g‘𝑅))
63	62	pm2.24d 146	. . . . . . . . . . . . 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) → (𝑓‘𝑥) = 0 ))
64	63	com12 32	. . . . . . . . . . . 12 ⊢ (¬ (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 ))
65		simplr 788	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑀 ∈ LMod)
66	65	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑀 ∈ LMod)
67	17	fveq2i 6106	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
68	22, 67	eqtr2i 2633	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(Scalar‘𝑀)) = 𝐵
69	68	oveq1i 6559	. . . . . . . . . . . . . . . . 17 ⊢ ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑥})) = (𝐵 ↑𝑚 (𝑆 ∖ {𝑥}))
70	45, 69	syl6eleqr 2699	. . . . . . . . . . . . . . . 16 ⊢ ((((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})))
71		ssdifss 3703	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
72	71	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
73	72	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
74		difexg 4735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
75	74	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → (𝑆 ∖ {𝑥}) ∈ V)
76	75	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
77		elpwg 4116	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∖ {𝑥}) ∈ V → ((𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
78	76, 77	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → ((𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
79	73, 78	mpbird 246	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
80	79	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((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))
81		lincval 41992	. . . . . . . . . . . . . . . 16 ⊢ ((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))))
82	66, 70, 80, 81	syl3anc 1318	. . . . . . . . . . . . . . 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}))( linC ‘𝑀)(𝑆 ∖ {𝑥})) = (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·𝑠 ‘𝑀)𝑧))))
83		fvres 6117	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧) = (𝑓‘𝑧))
84	83	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((((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))
85	84	oveq1d 6564	. . . . . . . . . . . . . . . . 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)( ·𝑠 ‘𝑀)𝑧) = ((𝑓‘𝑧)( ·𝑠 ‘𝑀)𝑧))
86	85	mpteq2dva 4672	. . . . . . . . . . . . . . . 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)) = (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑧)( ·𝑠 ‘𝑀)𝑧)))
87	86	oveq2d 6565	. . . . . . . . . . . . . . 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 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·𝑠 ‘𝑀)𝑧))) = (𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑧)( ·𝑠 ‘𝑀)𝑧))))
88		simplr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((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))
89		3anass 1035	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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	bicomi 213	. . . . . . . . . . . . . . . . . 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))
91	90	biimpi 205	. . . . . . . . . . . . . . . . 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))
92	91	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((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))
93	17, 22, 23, 41, 42, 43	lindslinindimp2lem4 42044	. . . . . . . . . . . . . . . 16 ⊢ (((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	36, 88, 92, 93	syl3anc 1318	. . . . . . . . . . . . . . 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))
95	82, 87, 94	3eqtrrd 2649	. . . . . . . . . . . . . 14 ⊢ ((((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})))
96	95	pm2.24d 146	. . . . . . . . . . . . 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})) → (𝑓‘𝑥) = 0 ))
97	96	com12 32	. . . . . . . . . . . 12 ⊢ (¬ (((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	64, 97	jaoi 393	. . . . . . . . . . 11 ⊢ ((¬ (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 ))
99	54, 98	syl 17	. . . . . . . . . 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})))) → ((((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	ex 449	. . . . . . . . 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 )))
101	100	com23 84	. . . . . . . 8 ⊢ ((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 )))
102	45, 101	mpcom 37	. . . . . . 7 ⊢ ((((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 ))
103	34, 102	syl5 33	. . . . . 6 ⊢ ((((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	103	expd 451	. . . . 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 )))
105	28, 104	syld 46	. . . 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 → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵 ↑𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓‘𝑥) = 0 )))
106	105	com12 32	. . 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 )))
107	106	expd 451	. 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 ))))
108	2, 107	pm2.61i 175	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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744   finSupp cfsupp 8158  Basecbs 15695  Scalarcsca 15771   ·_𝑠 cvsca 15772  0_gc0g 15923   Σ_g cgsu 15924  Grpcgrp 17245  inv_gcminusg 17246  LModclmod 18686   linC clinc 41987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-linc 41989

This theorem is referenced by:  lindslinindsimp2  42046