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Theorem lindslinindimp2lem4 44444
Description: Lemma 4 for lindslinindsimp2 44446. (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 767 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑆 ⊆ (Base‘𝑀))
4 elpwg 4541 . . . . . . . . . . . . . 14 (𝑆𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
54ad2antrr 722 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
63, 5mpbird 258 . . . . . . . . . . . 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 1120 . . . . . . . . . . 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 2818 . . . . . . . . . . 11 (Base‘𝑀) = (Base‘𝑀)
15 lindslinind.r . . . . . . . . . . 11 𝑅 = (Scalar‘𝑀)
16 lindslinind.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
17 eqid 2818 . . . . . . . . . . 11 ( ·𝑠𝑀) = ( ·𝑠𝑀)
18 eqid 2818 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
19 lindslinind.0 . . . . . . . . . . 11 0 = (0g𝑅)
2014, 15, 16, 17, 18, 19lincdifsn 44407 . . . . . . . . . 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 1363 . . . . . . . . 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 2820 . . . . . . . 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 19570 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
2423adantl 482 . . . . . . . . . 10 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ Grp)
2524ad2antrl 724 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑀 ∈ Grp)
261ad2antrl 724 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑀 ∈ LMod)
27 elmapi 8417 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐵m 𝑆) → 𝑓:𝑆𝐵)
28 ffvelrn 6841 . . . . . . . . . . . . . . 15 ((𝑓:𝑆𝐵𝑥𝑆) → (𝑓𝑥) ∈ 𝐵)
2928expcom 414 . . . . . . . . . . . . . 14 (𝑥𝑆 → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
3029ad2antll 725 . . . . . . . . . . . . 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 3959 . . . . . . . . . . 11 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝑀))
3534ad2antll 725 . . . . . . . . . 10 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑥 ∈ (Base‘𝑀))
3614, 15, 17, 16lmodvscl 19580 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ (𝑓𝑥) ∈ 𝐵𝑥 ∈ (Base‘𝑀)) → ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
3726, 33, 35, 36syl3anc 1363 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
38 difexg 5222 . . . . . . . . . . . . 13 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
3938ad2antrr 722 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
40 ssdifss 4109 . . . . . . . . . . . . 13 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
4140ad2antrl 724 . . . . . . . . . . . 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 767 . . . . . . . . . . . 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 725 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑆 ⊆ (Base‘𝑀))
477ad2antll 725 . . . . . . . . . . . 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 44442 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵m 𝑆))) → 𝐺 ∈ (𝐵m (𝑆 ∖ {𝑥})))
5344, 46, 47, 49, 52syl13anc 1364 . . . . . . . . . . 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 44443 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 )
5744, 55, 11, 56syl3anc 1363 . . . . . . . . . . 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 44396 . . . . . . . . . 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 1363 . . . . . . . . 9 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀))
61 eqid 2818 . . . . . . . . . 10 (invg𝑀) = (invg𝑀)
6214, 18, 50, 61grpinvid2 18093 . . . . . . . . 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 1363 . . . . . . . 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 283 . . . . . . 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 2825 . . . . . . . 8 (((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)))
6615fveq2i 6666 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
6716, 66eqtri 2841 . . . . . . . . . . . . 13 𝐵 = (Base‘(Scalar‘𝑀))
6867oveq1i 7155 . . . . . . . . . . . 12 (𝐵m (𝑆 ∖ {𝑥})) = ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥}))
6953, 68eleqtrdi 2920 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑m (𝑆 ∖ {𝑥})))
7039, 41elpwd 4546 . . . . . . . . . . . 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 44392 . . . . . . . . . . 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 1363 . . . . . . . . . 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 2820 . . . . . . . . 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 6664 . . . . . . . . . . . . . . . 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 6682 . . . . . . . . . . . . . . . 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 2853 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → (𝐺𝑦) = (𝑓𝑦))
8079oveq1d 7160 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝐺𝑦)( ·𝑠𝑀)𝑦) = ((𝑓𝑦)( ·𝑠𝑀)𝑦))
8180mpteq2dva 5152 . . . . . . . . . . . 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 7161 . . . . . . . . . . 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 2818 . . . . . . . . . . . . 13 (invg𝑅) = (invg𝑅)
8414, 15, 17, 61, 16, 83, 26, 35, 33lmodvsneg 19607 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
8551eqcomi 2827 . . . . . . . . . . . . . 14 ((invg𝑅)‘(𝑓𝑥)) = 𝑌
8685a1i 11 . . . . . . . . . . . . 13 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑅)‘(𝑓𝑥)) = 𝑌)
8786oveq1d 7160 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑌( ·𝑠𝑀)𝑥))
8884, 87eqtrd 2853 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵m 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝑌( ·𝑠𝑀)𝑥))
8982, 88eqeq12d 2834 . . . . . . . . . 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 230 . . . . . . . . 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 241 . . . . . . . 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 243 . . . . . . 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 241 . . . . . 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 1109 . . 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 1109 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  wss 3933  𝒫 cpw 4535  {csn 4557   class class class wbr 5057  cmpt 5137  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395   finSupp cfsupp 8821  Basecbs 16471  +gcplusg 16553  Scalarcsca 16556   ·𝑠 cvsca 16557  0gc0g 16701   Σg cgsu 16702  Grpcgrp 18041  invgcminusg 18042  LModclmod 19563   linC clinc 44387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-0g 16703  df-gsum 16704  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-grp 18044  df-minusg 18045  df-mulg 18163  df-cntz 18385  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-lmod 19565  df-linc 44389
This theorem is referenced by:  lindslinindsimp2lem5  44445
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