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Theorem lindslinindimp2lem4 46532
Description: Lemma 4 for lindslinindsimp2 46534. (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 485 . . . . . . . . . . . . 13 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ LMod)
21adantr 481 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑀 ∈ LMod)
3 simprl 769 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑆 ⊆ (Base‘𝑀))
4 elpwg 4563 . . . . . . . . . . . . . 14 (𝑆𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
54ad2antrr 724 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
63, 5mpbird 256 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑆 ∈ 𝒫 (Base‘𝑀))
7 simpr 485 . . . . . . . . . . . . 13 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑥𝑆)
87adantl 482 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑥𝑆)
92, 6, 83jca 1128 . . . . . . . . . . 11 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑆))
109adantl 482 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑆))
11 simpl 483 . . . . . . . . . 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 2736 . . . . . . . . . . 11 (Base‘𝑀) = (Base‘𝑀)
15 lindslinind.r . . . . . . . . . . 11 𝑅 = (Scalar‘𝑀)
16 lindslinind.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
17 eqid 2736 . . . . . . . . . . 11 ( ·𝑠𝑀) = ( ·𝑠𝑀)
18 eqid 2736 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
19 lindslinind.0 . . . . . . . . . . 11 0 = (0g𝑅)
2014, 15, 16, 17, 18, 19lincdifsn 46495 . . . . . . . . . 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 1371 . . . . . . . . 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 2738 . . . . . . . 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 20329 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
2423adantl 482 . . . . . . . . . 10 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ Grp)
2524ad2antrl 726 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑀 ∈ Grp)
261ad2antrl 726 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑀 ∈ LMod)
27 elmapi 8787 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐵m 𝑆) → 𝑓:𝑆𝐵)
28 ffvelcdm 7032 . . . . . . . . . . . . . . 15 ((𝑓:𝑆𝐵𝑥𝑆) → (𝑓𝑥) ∈ 𝐵)
2928expcom 414 . . . . . . . . . . . . . 14 (𝑥𝑆 → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
3029ad2antll 727 . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
3332imp 407 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑓𝑥) ∈ 𝐵)
34 ssel2 3939 . . . . . . . . . . 11 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝑀))
3534ad2antll 727 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑥 ∈ (Base‘𝑀))
3614, 15, 17, 16lmodvscl 20339 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ (𝑓𝑥) ∈ 𝐵𝑥 ∈ (Base‘𝑀)) → ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
3726, 33, 35, 36syl3anc 1371 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
38 difexg 5284 . . . . . . . . . . . . 13 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
3938ad2antrr 724 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
40 ssdifss 4095 . . . . . . . . . . . . 13 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
4140ad2antrl 726 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
4239, 41jca 512 . . . . . . . . . . 11 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
4342adantl 482 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
44 simprl 769 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆𝑉𝑀 ∈ LMod))
45 simpl 483 . . . . . . . . . . . . 13 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑆 ⊆ (Base‘𝑀))
4645ad2antll 727 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑆 ⊆ (Base‘𝑀))
477ad2antll 727 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑥𝑆)
48 simpl 483 . . . . . . . . . . . . 13 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) → 𝑓 ∈ (𝐵m 𝑆))
4948adantr 481 . . . . . . . . . . . 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 46530 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵m 𝑆))) → 𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})))
5344, 46, 47, 49, 52syl13anc 1372 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})))
54 simpr 485 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
5554adantl 482 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
5615, 16, 19, 50, 51, 12lindslinindimp2lem3 46531 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 )
5744, 55, 11, 56syl3anc 1371 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 finSupp 0 )
5853, 57jca 512 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})) ∧ 𝐺 finSupp 0 ))
5914, 15, 16, 19lincfsuppcl 46484 . . . . . . . . . 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 1371 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀))
61 eqid 2736 . . . . . . . . . 10 (invg𝑀) = (invg𝑀)
6214, 18, 50, 61grpinvid2 18803 . . . . . . . . 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 1371 . . . . . . . 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 281 . . . . . . 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 2743 . . . . . . . 8 (((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)))
6615fveq2i 6845 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
6716, 66eqtri 2764 . . . . . . . . . . . . 13 𝐵 = (Base‘(Scalar‘𝑀))
6867oveq1i 7367 . . . . . . . . . . . 12 (𝐵m (𝑆 ∖ {𝑥})) = ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥}))
6953, 68eleqtrdi 2848 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥})))
7039, 41elpwd 4566 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
7170adantl 482 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
72 lincval 46480 . . . . . . . . . . 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 1371 . . . . . . . . . 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 2738 . . . . . . . . 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 6843 . . . . . . . . . . . . . . . 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 6861 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓𝑦))
7877adantl 482 . . . . . . . . . . . . . . 15 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓𝑦))
7976, 78eqtrd 2776 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → (𝐺𝑦) = (𝑓𝑦))
8079oveq1d 7372 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝐺𝑦)( ·𝑠𝑀)𝑦) = ((𝑓𝑦)( ·𝑠𝑀)𝑦))
8180mpteq2dva 5205 . . . . . . . . . . . 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 7373 . . . . . . . . . . 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 2736 . . . . . . . . . . . . 13 (invg𝑅) = (invg𝑅)
8414, 15, 17, 61, 16, 83, 26, 35, 33lmodvsneg 20366 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
8551eqcomi 2745 . . . . . . . . . . . . . 14 ((invg𝑅)‘(𝑓𝑥)) = 𝑌
8685a1i 11 . . . . . . . . . . . . 13 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑅)‘(𝑓𝑥)) = 𝑌)
8786oveq1d 7372 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑌( ·𝑠𝑀)𝑥))
8884, 87eqtrd 2776 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝑌( ·𝑠𝑀)𝑥))
8982, 88eqeq12d 2752 . . . . . . . . . 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 228 . . . . . . . . 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 239 . . . . . . . 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 241 . . . . . . 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 239 . . . . . 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 413 . . . . 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 1117 . . 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 1117 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  wss 3910  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cmpt 5188  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765   finSupp cfsupp 9305  Basecbs 17083  +gcplusg 17133  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321   Σg cgsu 17322  Grpcgrp 18748  invgcminusg 18749  LModclmod 20322   linC clinc 46475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-lmod 20324  df-linc 46477
This theorem is referenced by:  lindslinindsimp2lem5  46533
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