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Theorem lindslinindimp2lem4 48378
Description: Lemma 4 for lindslinindsimp2 48380. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.)
Hypotheses
Ref Expression
lindslinind.r 𝑅 = (Scalar‘𝑀)
lindslinind.b 𝐵 = (Base‘𝑅)
lindslinind.0 0 = (0g𝑅)
lindslinind.z 𝑍 = (0g𝑀)
lindslinind.y 𝑌 = ((invg𝑅)‘(𝑓𝑥))
lindslinind.g 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))
Assertion
Ref Expression
lindslinindimp2lem4 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥))
Distinct variable groups:   𝐵,𝑓,𝑦   𝑓,𝑀,𝑦   𝑅,𝑓,𝑥   𝑆,𝑓,𝑥,𝑦   𝑦,𝑉   𝑓,𝑍,𝑦   0 ,𝑓,𝑥,𝑦   𝑦,𝐺
Allowed substitution hints:   𝐵(𝑥)   𝑅(𝑦)   𝐺(𝑥,𝑓)   𝑀(𝑥)   𝑉(𝑥,𝑓)   𝑌(𝑥,𝑦,𝑓)   𝑍(𝑥)

Proof of Theorem lindslinindimp2lem4
StepHypRef Expression
1 simpr 484 . . . . . . . . . . . . 13 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ LMod)
21adantr 480 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑀 ∈ LMod)
3 simprl 771 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑆 ⊆ (Base‘𝑀))
4 elpwg 4603 . . . . . . . . . . . . . 14 (𝑆𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
54ad2antrr 726 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
63, 5mpbird 257 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑆 ∈ 𝒫 (Base‘𝑀))
7 simpr 484 . . . . . . . . . . . . 13 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑥𝑆)
87adantl 481 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑥𝑆)
92, 6, 83jca 1129 . . . . . . . . . . 11 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑆))
109adantl 481 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑆))
11 simpl 482 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ))
12 lindslinind.g . . . . . . . . . . 11 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))
1312a1i 11 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥})))
14 eqid 2737 . . . . . . . . . . 11 (Base‘𝑀) = (Base‘𝑀)
15 lindslinind.r . . . . . . . . . . 11 𝑅 = (Scalar‘𝑀)
16 lindslinind.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
17 eqid 2737 . . . . . . . . . . 11 ( ·𝑠𝑀) = ( ·𝑠𝑀)
18 eqid 2737 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
19 lindslinind.0 . . . . . . . . . . 11 0 = (0g𝑅)
2014, 15, 16, 17, 18, 19lincdifsn 48341 . . . . . . . . . 10 (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))) → (𝑓( linC ‘𝑀)𝑆) = ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))(+g𝑀)((𝑓𝑥)( ·𝑠𝑀)𝑥)))
2110, 11, 13, 20syl3anc 1373 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑓( linC ‘𝑀)𝑆) = ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))(+g𝑀)((𝑓𝑥)( ·𝑠𝑀)𝑥)))
2221eqeq1d 2739 . . . . . . . 8 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))(+g𝑀)((𝑓𝑥)( ·𝑠𝑀)𝑥)) = 𝑍))
23 lmodgrp 20865 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
2423adantl 481 . . . . . . . . . 10 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ Grp)
2524ad2antrl 728 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑀 ∈ Grp)
261ad2antrl 728 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑀 ∈ LMod)
27 elmapi 8889 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐵m 𝑆) → 𝑓:𝑆𝐵)
28 ffvelcdm 7101 . . . . . . . . . . . . . . 15 ((𝑓:𝑆𝐵𝑥𝑆) → (𝑓𝑥) ∈ 𝐵)
2928expcom 413 . . . . . . . . . . . . . 14 (𝑥𝑆 → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
3029ad2antll 729 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
3127, 30syl5com 31 . . . . . . . . . . . 12 (𝑓 ∈ (𝐵m 𝑆) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
3231adantr 480 . . . . . . . . . . 11 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
3332imp 406 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑓𝑥) ∈ 𝐵)
34 ssel2 3978 . . . . . . . . . . 11 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝑀))
3534ad2antll 729 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑥 ∈ (Base‘𝑀))
3614, 15, 17, 16lmodvscl 20876 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ (𝑓𝑥) ∈ 𝐵𝑥 ∈ (Base‘𝑀)) → ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
3726, 33, 35, 36syl3anc 1373 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
38 difexg 5329 . . . . . . . . . . . . 13 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
3938ad2antrr 726 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
40 ssdifss 4140 . . . . . . . . . . . . 13 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
4140ad2antrl 728 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
4239, 41jca 511 . . . . . . . . . . 11 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
4342adantl 481 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
44 simprl 771 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆𝑉𝑀 ∈ LMod))
45 simpl 482 . . . . . . . . . . . . 13 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑆 ⊆ (Base‘𝑀))
4645ad2antll 729 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑆 ⊆ (Base‘𝑀))
477ad2antll 729 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑥𝑆)
48 simpl 482 . . . . . . . . . . . . 13 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) → 𝑓 ∈ (𝐵m 𝑆))
4948adantr 480 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑓 ∈ (𝐵m 𝑆))
50 lindslinind.z . . . . . . . . . . . . 13 𝑍 = (0g𝑀)
51 lindslinind.y . . . . . . . . . . . . 13 𝑌 = ((invg𝑅)‘(𝑓𝑥))
5215, 16, 19, 50, 51, 12lindslinindimp2lem2 48376 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵m 𝑆))) → 𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})))
5344, 46, 47, 49, 52syl13anc 1374 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})))
54 simpr 484 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
5554adantl 481 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
5615, 16, 19, 50, 51, 12lindslinindimp2lem3 48377 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 )
5744, 55, 11, 56syl3anc 1373 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 finSupp 0 )
5853, 57jca 511 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})) ∧ 𝐺 finSupp 0 ))
5914, 15, 16, 19lincfsuppcl 48330 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)) ∧ (𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})) ∧ 𝐺 finSupp 0 )) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀))
6026, 43, 58, 59syl3anc 1373 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀))
61 eqid 2737 . . . . . . . . . 10 (invg𝑀) = (invg𝑀)
6214, 18, 50, 61grpinvid2 19010 . . . . . . . . 9 ((𝑀 ∈ Grp ∧ ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀) ∧ (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀)) → (((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))(+g𝑀)((𝑓𝑥)( ·𝑠𝑀)𝑥)) = 𝑍))
6325, 37, 60, 62syl3anc 1373 . . . . . . . 8 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))(+g𝑀)((𝑓𝑥)( ·𝑠𝑀)𝑥)) = 𝑍))
6422, 63bitr4d 282 . . . . . . 7 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥}))))
65 eqcom 2744 . . . . . . . 8 (((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)))
6615fveq2i 6909 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
6716, 66eqtri 2765 . . . . . . . . . . . . 13 𝐵 = (Base‘(Scalar‘𝑀))
6867oveq1i 7441 . . . . . . . . . . . 12 (𝐵m (𝑆 ∖ {𝑥})) = ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥}))
6953, 68eleqtrdi 2851 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥})))
7039, 41elpwd 4606 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
7170adantl 481 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
72 lincval 48326 . . . . . . . . . . 11 ((𝑀 ∈ LMod ∧ 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥})) ∧ (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀)) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺𝑦)( ·𝑠𝑀)𝑦))))
7326, 69, 71, 72syl3anc 1373 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺𝑦)( ·𝑠𝑀)𝑦))))
7473eqeq1d 2739 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) ↔ (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺𝑦)( ·𝑠𝑀)𝑦))) = ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥))))
7512fveq1i 6907 . . . . . . . . . . . . . . . 16 (𝐺𝑦) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦)
7675a1i 11 . . . . . . . . . . . . . . 15 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → (𝐺𝑦) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦))
77 fvres 6925 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓𝑦))
7877adantl 481 . . . . . . . . . . . . . . 15 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓𝑦))
7976, 78eqtrd 2777 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → (𝐺𝑦) = (𝑓𝑦))
8079oveq1d 7446 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝐺𝑦)( ·𝑠𝑀)𝑦) = ((𝑓𝑦)( ·𝑠𝑀)𝑦))
8180mpteq2dva 5242 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺𝑦)( ·𝑠𝑀)𝑦)) = (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦)))
8281oveq2d 7447 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺𝑦)( ·𝑠𝑀)𝑦))) = (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))))
83 eqid 2737 . . . . . . . . . . . . 13 (invg𝑅) = (invg𝑅)
8414, 15, 17, 61, 16, 83, 26, 35, 33lmodvsneg 20904 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
8551eqcomi 2746 . . . . . . . . . . . . . 14 ((invg𝑅)‘(𝑓𝑥)) = 𝑌
8685a1i 11 . . . . . . . . . . . . 13 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑅)‘(𝑓𝑥)) = 𝑌)
8786oveq1d 7446 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑌( ·𝑠𝑀)𝑥))
8884, 87eqtrd 2777 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝑌( ·𝑠𝑀)𝑥))
8982, 88eqeq12d 2753 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺𝑦)( ·𝑠𝑀)𝑦))) = ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) ↔ (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥)))
9089biimpd 229 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝐺𝑦)( ·𝑠𝑀)𝑦))) = ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥)))
9174, 90sylbid 240 . . . . . . . 8 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥)))
9265, 91biimtrid 242 . . . . . . 7 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥)))
9364, 92sylbid 240 . . . . . 6 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥)))
9493ex 412 . . . . 5 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥))))
9594com23 86 . . . 4 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥))))
96953impia 1118 . . 3 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥)))
9796com12 32 . 2 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥)))
98973impia 1118 1 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  wss 3951  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  cmpt 5225  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866   finSupp cfsupp 9401  Basecbs 17247  +gcplusg 17297  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484   Σg cgsu 17485  Grpcgrp 18951  invgcminusg 18952  LModclmod 20858   linC clinc 48321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-lmod 20860  df-linc 48323
This theorem is referenced by:  lindslinindsimp2lem5  48379
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