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Theorem lindslinindimp2lem4 44688
Description: Lemma 4 for lindslinindsimp2 44690. (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 488 . . . . . . . . . . . . 13 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ LMod)
21adantr 484 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑀 ∈ LMod)
3 simprl 770 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑆 ⊆ (Base‘𝑀))
4 elpwg 4515 . . . . . . . . . . . . . 14 (𝑆𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
54ad2antrr 725 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
63, 5mpbird 260 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑆 ∈ 𝒫 (Base‘𝑀))
7 simpr 488 . . . . . . . . . . . . 13 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑥𝑆)
87adantl 485 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑥𝑆)
92, 6, 83jca 1125 . . . . . . . . . . 11 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑆))
109adantl 485 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑆))
11 simpl 486 . . . . . . . . . 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 2821 . . . . . . . . . . 11 (Base‘𝑀) = (Base‘𝑀)
15 lindslinind.r . . . . . . . . . . 11 𝑅 = (Scalar‘𝑀)
16 lindslinind.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
17 eqid 2821 . . . . . . . . . . 11 ( ·𝑠𝑀) = ( ·𝑠𝑀)
18 eqid 2821 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
19 lindslinind.0 . . . . . . . . . . 11 0 = (0g𝑅)
2014, 15, 16, 17, 18, 19lincdifsn 44651 . . . . . . . . . 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 1368 . . . . . . . . 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 2823 . . . . . . . 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 19617 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
2423adantl 485 . . . . . . . . . 10 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ Grp)
2524ad2antrl 727 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑀 ∈ Grp)
261ad2antrl 727 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑀 ∈ LMod)
27 elmapi 8403 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐵m 𝑆) → 𝑓:𝑆𝐵)
28 ffvelrn 6822 . . . . . . . . . . . . . . 15 ((𝑓:𝑆𝐵𝑥𝑆) → (𝑓𝑥) ∈ 𝐵)
2928expcom 417 . . . . . . . . . . . . . 14 (𝑥𝑆 → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
3029ad2antll 728 . . . . . . . . . . . . 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 484 . . . . . . . . . . 11 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
3332imp 410 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑓𝑥) ∈ 𝐵)
34 ssel2 3938 . . . . . . . . . . 11 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝑀))
3534ad2antll 728 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑥 ∈ (Base‘𝑀))
3614, 15, 17, 16lmodvscl 19627 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ (𝑓𝑥) ∈ 𝐵𝑥 ∈ (Base‘𝑀)) → ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
3726, 33, 35, 36syl3anc 1368 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
38 difexg 5204 . . . . . . . . . . . . 13 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
3938ad2antrr 725 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
40 ssdifss 4088 . . . . . . . . . . . . 13 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
4140ad2antrl 727 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
4239, 41jca 515 . . . . . . . . . . 11 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
4342adantl 485 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
44 simprl 770 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆𝑉𝑀 ∈ LMod))
45 simpl 486 . . . . . . . . . . . . 13 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑆 ⊆ (Base‘𝑀))
4645ad2antll 728 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑆 ⊆ (Base‘𝑀))
477ad2antll 728 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑥𝑆)
48 simpl 486 . . . . . . . . . . . . 13 ((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) → 𝑓 ∈ (𝐵m 𝑆))
4948adantr 484 . . . . . . . . . . . 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 44686 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵m 𝑆))) → 𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})))
5344, 46, 47, 49, 52syl13anc 1369 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})))
54 simpr 488 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
5554adantl 485 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
5615, 16, 19, 50, 51, 12lindslinindimp2lem3 44687 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 )
5744, 55, 11, 56syl3anc 1368 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 finSupp 0 )
5853, 57jca 515 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})) ∧ 𝐺 finSupp 0 ))
5914, 15, 16, 19lincfsuppcl 44640 . . . . . . . . . 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 1368 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀))
61 eqid 2821 . . . . . . . . . 10 (invg𝑀) = (invg𝑀)
6214, 18, 50, 61grpinvid2 18134 . . . . . . . . 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 1368 . . . . . . . 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 285 . . . . . . 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 2828 . . . . . . . 8 (((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)))
6615fveq2i 6646 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
6716, 66eqtri 2844 . . . . . . . . . . . . 13 𝐵 = (Base‘(Scalar‘𝑀))
6867oveq1i 7140 . . . . . . . . . . . 12 (𝐵m (𝑆 ∖ {𝑥})) = ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥}))
6953, 68eleqtrdi 2922 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥})))
7039, 41elpwd 4520 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
7170adantl 485 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
72 lincval 44636 . . . . . . . . . . 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 1368 . . . . . . . . . 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 2823 . . . . . . . . 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 6644 . . . . . . . . . . . . . . . 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 6662 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓𝑦))
7877adantl 485 . . . . . . . . . . . . . . 15 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓𝑦))
7976, 78eqtrd 2856 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → (𝐺𝑦) = (𝑓𝑦))
8079oveq1d 7145 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝐺𝑦)( ·𝑠𝑀)𝑦) = ((𝑓𝑦)( ·𝑠𝑀)𝑦))
8180mpteq2dva 5134 . . . . . . . . . . . 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 7146 . . . . . . . . . . 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 2821 . . . . . . . . . . . . 13 (invg𝑅) = (invg𝑅)
8414, 15, 17, 61, 16, 83, 26, 35, 33lmodvsneg 19654 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
8551eqcomi 2830 . . . . . . . . . . . . . 14 ((invg𝑅)‘(𝑓𝑥)) = 𝑌
8685a1i 11 . . . . . . . . . . . . 13 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑅)‘(𝑓𝑥)) = 𝑌)
8786oveq1d 7145 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑌( ·𝑠𝑀)𝑥))
8884, 87eqtrd 2856 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝑌( ·𝑠𝑀)𝑥))
8982, 88eqeq12d 2837 . . . . . . . . . 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 232 . . . . . . . . 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 243 . . . . . . . 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, 91syl5bi 245 . . . . . . 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 243 . . . . . 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 416 . . . . 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 1114 . . 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 1114 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3471  cdif 3907  wss 3910  𝒫 cpw 4512  {csn 4540   class class class wbr 5039  cmpt 5119  cres 5530  wf 6324  cfv 6328  (class class class)co 7130  m cmap 8381   finSupp cfsupp 8809  Basecbs 16462  +gcplusg 16544  Scalarcsca 16547   ·𝑠 cvsca 16548  0gc0g 16692   Σg cgsu 16693  Grpcgrp 18082  invgcminusg 18083  LModclmod 19610   linC clinc 44631
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-of 7384  df-om 7556  df-1st 7664  df-2nd 7665  df-supp 7806  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-map 8383  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-fsupp 8810  df-oi 8950  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-n0 11876  df-z 11960  df-uz 12222  df-fz 12876  df-fzo 13017  df-seq 13353  df-hash 13675  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-plusg 16557  df-0g 16694  df-gsum 16695  df-mre 16836  df-mrc 16837  df-acs 16839  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-submnd 17936  df-grp 18085  df-minusg 18086  df-mulg 18204  df-cntz 18426  df-cmn 18887  df-abl 18888  df-mgp 19219  df-ur 19231  df-ring 19278  df-lmod 19612  df-linc 44633
This theorem is referenced by:  lindslinindsimp2lem5  44689
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