lindslinindsimp2lem5 - Mathbox for Alexander van der Vekens

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Theorem lindslinindsimp2lem5 47642

Description: Lemma 5 for lindslinindsimp2 47643. (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 = (0_g‘𝑅)
lindslinind.z	⊢ 𝑍 = (0_g‘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 8866	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐵 ↑_m 𝑆) → 𝑓:𝑆⟶𝐵)
4		ffvelcdm 7088	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝑆⟶𝐵 ∧ 𝑥 ∈ 𝑆) → (𝑓‘𝑥) ∈ 𝐵)
5	4	expcom 412	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑆 → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
6	5	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → (𝑓:𝑆⟶𝐵 → (𝑓‘𝑥) ∈ 𝐵))
7	6	adantl 480	. . . . . . . . . . 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 479	. . . . . . . 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 406	. . . . . . 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 531	. . . . . 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 2931	. . . . . . 7 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) = 0 )
14	13	bicomi 223	. . . . . 6 ⊢ (¬ (𝑓‘𝑥) = 0 ↔ (𝑓‘𝑥) ≠ 0 )
15		eldifsn 4791	. . . . . 6 ⊢ ((𝑓‘𝑥) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ≠ 0 ))
16	12, 14, 15	3bitr4g 313	. . . . 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 20756	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp)
19	18	adantl 480	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑅 ∈ Grp)
20	19	adantr 479	. . . . . . . 8 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑅 ∈ Grp)
21	20	adantr 479	. . . . . . 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 = (0_g‘𝑅)
24		eqid 2725	. . . . . . . 8 ⊢ (inv_g‘𝑅) = (inv_g‘𝑅)
25	22, 23, 24	grpinvnzcl 18971	. . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ (𝑓‘𝑥) ∈ (𝐵 ∖ { 0 })) → ((inv_g‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 }))
26	21, 25	sylan 578	. . . . . 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 })) → ((inv_g‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 }))
27	26	ex 411	. . . . 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 }) → ((inv_g‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 })))
28	16, 27	sylbid 239	. . . 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 → ((inv_g‘𝑅)‘(𝑓‘𝑥)) ∈ (𝐵 ∖ { 0 })))
29		oveq1 7424	. . . . . . . . . . 11 ⊢ (𝑦 = ((inv_g‘𝑅)‘(𝑓‘𝑥)) → (𝑦( ·_𝑠 ‘𝑀)𝑥) = (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥))
30	29	eqeq1d 2727	. . . . . . . . . 10 ⊢ (𝑦 = ((inv_g‘𝑅)‘(𝑓‘𝑥)) → ((𝑦( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
31	30	notbid 317	. . . . . . . . 9 ⊢ (𝑦 = ((inv_g‘𝑅)‘(𝑓‘𝑥)) → (¬ (𝑦( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
32	31	orbi2d 913	. . . . . . . 8 ⊢ (𝑦 = ((inv_g‘𝑅)‘(𝑓‘𝑥)) → ((¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) ↔ (¬ 𝑔 finSupp 0 ∨ ¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
33	32	ralbidv 3168	. . . . . . 7 ⊢ (𝑦 = ((inv_g‘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 ∨ ¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
34	33	rspcva 3605	. . . . . 6 ⊢ ((((inv_g‘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 ∨ ¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
35		simpl 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod))
36	35	adantr 479	. . . . . . . 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 775	. . . . . . . 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 776	. . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 ∈ (𝐵 ↑_m 𝑆))
40	39	adantl 480	. . . . . . . 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 = (0_g‘𝑀)
42		eqid 2725	. . . . . . . . 9 ⊢ ((inv_g‘𝑅)‘(𝑓‘𝑥)) = ((inv_g‘𝑅)‘(𝑓‘𝑥))
43		eqid 2725	. . . . . . . . 9 ⊢ (𝑓 ↾ (𝑆 ∖ {𝑥})) = (𝑓 ↾ (𝑆 ∖ {𝑥}))
44	17, 22, 23, 41, 42, 43	lindslinindimp2lem2 47639	. . . . . . . 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 1369	. . . . . . 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 = (0_g‘𝑅))
48	46, 47	breq12d 5161	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔 finSupp 0 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0_g‘𝑅)))
49	48	notbid 317	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ 𝑔 finSupp 0 ↔ ¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0_g‘𝑅)))
50		oveq1 7424	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))
51	50	eqeq2d 2736	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → ((((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
52	51	notbid 317	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → (¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
53	49, 52	orbi12d 916	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑓 ↾ (𝑆 ∖ {𝑥})) → ((¬ 𝑔 finSupp 0 ∨ ¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) ↔ (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0_g‘𝑅) ∨ ¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})))))
54	53	rspcva 3605	. . . . . . . . . 10 ⊢ (((𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵 ↑_m (𝑆 ∖ {𝑥})) ∧ ∀𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥})))) → (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0_g‘𝑅) ∨ ¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
55	23	breq2i 5156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 finSupp 0 ↔ 𝑓 finSupp (0_g‘𝑅))
56	55	biimpi 215	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 finSupp 0 → 𝑓 finSupp (0_g‘𝑅))
57	56	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp (0_g‘𝑅))
58	57	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → 𝑓 finSupp (0_g‘𝑅))
59	58	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → 𝑓 finSupp (0_g‘𝑅))
60		fvexd 6909	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (0_g‘𝑅) ∈ V)
61	59, 60	fsuppres 9416	. . . . . . . . . . . . 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 (0_g‘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 (0_g‘𝑅) → (𝑓‘𝑥) = 0 ))
63	62	com12 32	. . . . . . . . . . 11 ⊢ (¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0_g‘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 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → 𝑀 ∈ LMod)
65	64	adantr 479	. . . . . . . . . . . . . . 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 6897	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝑅) = (Base‘(Scalar‘𝑀))
67	22, 66	eqtr2i 2754	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘𝑀)) = 𝐵
68	67	oveq1i 7427	. . . . . . . . . . . . . . . 16 ⊢ ((Base‘(Scalar‘𝑀)) ↑_m (𝑆 ∖ {𝑥})) = (𝐵 ↑_m (𝑆 ∖ {𝑥}))
69	45, 68	eleqtrrdi 2836	. . . . . . . . . . . . . . 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 4133	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
71	70	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
72	71	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
73		difexg 5329	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
74	73	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → (𝑆 ∖ {𝑥}) ∈ V)
75	74	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
76		elpwg 4606	. . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
79	78	adantr 479	. . . . . . . . . . . . . . 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 47589	. . . . . . . . . . . . . . 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 Σ_g (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·_𝑠 ‘𝑀)𝑧))))
81	65, 69, 79, 80	syl3anc 1368	. . . . . . . . . . . . . 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 Σ_g (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·_𝑠 ‘𝑀)𝑧))))
82		fvres 6913	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧) = (𝑓‘𝑧))
83	82	adantl 480	. . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . 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 5248	. . . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑀 Σ_g (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ (((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑧)( ·_𝑠 ‘𝑀)𝑧))) = (𝑀 Σ_g (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑧)( ·_𝑠 ‘𝑀)𝑧))))
87		simplr 767	. . . . . . . . . . . . . . 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 1092	. . . . . . . . . . . . . . . . . 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 223	. . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 47641	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σ_g (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑧)( ·_𝑠 ‘𝑀)𝑧))) = (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥))
93	36, 87, 91, 92	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (𝑀 Σ_g (𝑧 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓‘𝑧)( ·_𝑠 ‘𝑀)𝑧))) = (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥))
94	81, 86, 93	3eqtrrd 2770	. . . . . . . . . . . . 13 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (((inv_g‘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))) → (¬ (((inv_g‘𝑅)‘(𝑓‘𝑥))( ·_𝑠 ‘𝑀)𝑥) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))( linC ‘𝑀)(𝑆 ∖ {𝑥})) → (𝑓‘𝑥) = 0 ))
96	95	com12 32	. . . . . . . . . . 11 ⊢ (¬ (((inv_g‘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 855	. . . . . . . . . 10 ⊢ ((¬ (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp (0_g‘𝑅) ∨ ¬ (((inv_g‘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 ∨ ¬ (((inv_g‘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 411	. . . . . . . 8 ⊢ ((𝑓 ↾ (𝑆 ∖ {𝑥})) ∈ (𝐵 ↑_m (𝑆 ∖ {𝑥})) → (∀𝑔 ∈ (𝐵 ↑_m (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (((inv_g‘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 ∨ ¬ (((inv_g‘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 ∨ ¬ (((inv_g‘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))) → ((((inv_g‘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 414	. . . 4 ⊢ ((((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆)) ∧ (𝑓 ∈ (𝐵 ↑_m 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍))) → (((inv_g‘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 414	. 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 )))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 Vcvv 3463 ∖ cdif 3942 ⊆ wss 3945 𝒫 cpw 4603 {csn 4629 class class class wbr 5148 ↦ cmpt 5231 ↾ cres 5679 ⟶wf 6543 ‘cfv 6547 (class class class)co 7417 ↑_m cmap 8843 finSupp cfsupp 9385 Basecbs 17179 Scalarcsca 17235 ·_𝑠 cvsca 17236 0_gc0g 17420 Σ_g cgsu 17421 Grpcgrp 18894 inv_gcminusg 18895 LModclmod 20747 linC clinc 47584

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-om 7870 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-0g 17422 df-gsum 17423 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18897 df-minusg 18898 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-lmod 20749 df-linc 47586

This theorem is referenced by: lindslinindsimp2 47643

W3C validator