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Theorem lindslinindimp2lem4 42919
Description: Lemma 4 for lindslinindsimp2 42921. (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 = (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 477 . . . . . . . . . . . . 13 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ LMod)
21adantr 472 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑀 ∈ LMod)
3 simprl 787 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑆 ⊆ (Base‘𝑀))
4 elpwg 4323 . . . . . . . . . . . . . 14 (𝑆𝑉 → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
54ad2antrr 717 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
63, 5mpbird 248 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑆 ∈ 𝒫 (Base‘𝑀))
7 simpr 477 . . . . . . . . . . . . 13 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑥𝑆)
87adantl 473 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → 𝑥𝑆)
92, 6, 83jca 1158 . . . . . . . . . . 11 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑆))
109adantl 473 . . . . . . . . . 10 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥𝑆))
11 simpl 474 . . . . . . . . . 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 2765 . . . . . . . . . . 11 (Base‘𝑀) = (Base‘𝑀)
15 lindslinind.r . . . . . . . . . . 11 𝑅 = (Scalar‘𝑀)
16 lindslinind.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
17 eqid 2765 . . . . . . . . . . 11 ( ·𝑠𝑀) = ( ·𝑠𝑀)
18 eqid 2765 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
19 lindslinind.0 . . . . . . . . . . 11 0 = (0g𝑅)
2014, 15, 16, 17, 18, 19lincdifsn 42882 . . . . . . . . . 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 1490 . . . . . . . . 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 2767 . . . . . . . 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 19139 . . . . . . . . . . 11 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
2423adantl 473 . . . . . . . . . 10 ((𝑆𝑉𝑀 ∈ LMod) → 𝑀 ∈ Grp)
2524ad2antrl 719 . . . . . . . . 9 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑀 ∈ Grp)
261ad2antrl 719 . . . . . . . . . 10 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑀 ∈ LMod)
27 elmapi 8082 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐵𝑚 𝑆) → 𝑓:𝑆𝐵)
28 ffvelrn 6547 . . . . . . . . . . . . . . . 16 ((𝑓:𝑆𝐵𝑥𝑆) → (𝑓𝑥) ∈ 𝐵)
2928expcom 402 . . . . . . . . . . . . . . 15 (𝑥𝑆 → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
3029ad2antll 720 . . . . . . . . . . . . . 14 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓:𝑆𝐵 → (𝑓𝑥) ∈ 𝐵))
3130com12 32 . . . . . . . . . . . . 13 (𝑓:𝑆𝐵 → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
3227, 31syl 17 . . . . . . . . . . . 12 (𝑓 ∈ (𝐵𝑚 𝑆) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
3332adantr 472 . . . . . . . . . . 11 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
3433imp 395 . . . . . . . . . 10 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑓𝑥) ∈ 𝐵)
35 ssel2 3756 . . . . . . . . . . 11 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑥 ∈ (Base‘𝑀))
3635ad2antll 720 . . . . . . . . . 10 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑥 ∈ (Base‘𝑀))
3714, 15, 17, 16lmodvscl 19149 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ (𝑓𝑥) ∈ 𝐵𝑥 ∈ (Base‘𝑀)) → ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
3826, 34, 36, 37syl3anc 1490 . . . . . . . . 9 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑓𝑥)( ·𝑠𝑀)𝑥) ∈ (Base‘𝑀))
39 difexg 4969 . . . . . . . . . . . . 13 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
4039ad2antrr 717 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ V)
41 ssdifss 3903 . . . . . . . . . . . . 13 (𝑆 ⊆ (Base‘𝑀) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
4241ad2antrl 719 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀))
4340, 42jca 507 . . . . . . . . . . 11 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
4443adantl 473 . . . . . . . . . 10 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
45 simprl 787 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆𝑉𝑀 ∈ LMod))
46 simpl 474 . . . . . . . . . . . . 13 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) → 𝑆 ⊆ (Base‘𝑀))
4746ad2antll 720 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑆 ⊆ (Base‘𝑀))
487ad2antll 720 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑥𝑆)
49 simpl 474 . . . . . . . . . . . . 13 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) → 𝑓 ∈ (𝐵𝑚 𝑆))
5049adantr 472 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝑓 ∈ (𝐵𝑚 𝑆))
51 lindslinind.z . . . . . . . . . . . . 13 𝑍 = (0g𝑀)
52 lindslinind.y . . . . . . . . . . . . 13 𝑌 = ((invg𝑅)‘(𝑓𝑥))
5315, 16, 19, 51, 52, 12lindslinindimp2lem2 42917 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})))
5445, 47, 48, 50, 53syl13anc 1491 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})))
55 simpr 477 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
5655adantl 473 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))
5715, 16, 19, 51, 52, 12lindslinindimp2lem3 42918 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 )
5845, 56, 11, 57syl3anc 1490 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 finSupp 0 )
5954, 58jca 507 . . . . . . . . . 10 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})) ∧ 𝐺 finSupp 0 ))
6014, 15, 16, 19lincfsuppcl 42871 . . . . . . . . . 10 ((𝑀 ∈ LMod ∧ ((𝑆 ∖ {𝑥}) ∈ V ∧ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)) ∧ (𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})) ∧ 𝐺 finSupp 0 )) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀))
6126, 44, 59, 60syl3anc 1490 . . . . . . . . 9 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ∈ (Base‘𝑀))
62 eqid 2765 . . . . . . . . . 10 (invg𝑀) = (invg𝑀)
6314, 18, 51, 62grpinvid2 17738 . . . . . . . . 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))
6425, 38, 61, 63syl3anc 1490 . . . . . . . 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))
6522, 64bitr4d 273 . . . . . . 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}))))
66 eqcom 2772 . . . . . . . 8 (((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) ↔ (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑥})) = ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)))
6715fveq2i 6378 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘(Scalar‘𝑀))
6816, 67eqtri 2787 . . . . . . . . . . . . 13 𝐵 = (Base‘(Scalar‘𝑀))
6968oveq1i 6852 . . . . . . . . . . . 12 (𝐵𝑚 (𝑆 ∖ {𝑥})) = ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑥}))
7054, 69syl6eleq 2854 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → 𝐺 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 (𝑆 ∖ {𝑥})))
71 elpwg 4323 . . . . . . . . . . . . . 14 ((𝑆 ∖ {𝑥}) ∈ V → ((𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
7240, 71syl 17 . . . . . . . . . . . . 13 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀) ↔ (𝑆 ∖ {𝑥}) ⊆ (Base‘𝑀)))
7342, 72mpbird 248 . . . . . . . . . . . 12 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
7473adantl 473 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑆 ∖ {𝑥}) ∈ 𝒫 (Base‘𝑀))
75 lincval 42867 . . . . . . . . . . 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))))
7626, 70, 74, 75syl3anc 1490 . . . . . . . . . 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))))
7776eqeq1d 2767 . . . . . . . . 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))))
7812fveq1i 6376 . . . . . . . . . . . . . . . 16 (𝐺𝑦) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦)
7978a1i 11 . . . . . . . . . . . . . . 15 ((((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → (𝐺𝑦) = ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦))
80 fvres 6394 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (𝑆 ∖ {𝑥}) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓𝑦))
8180adantl 473 . . . . . . . . . . . . . . 15 ((((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝑓 ↾ (𝑆 ∖ {𝑥}))‘𝑦) = (𝑓𝑦))
8279, 81eqtrd 2799 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → (𝐺𝑦) = (𝑓𝑦))
8382oveq1d 6857 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) ∧ 𝑦 ∈ (𝑆 ∖ {𝑥})) → ((𝐺𝑦)( ·𝑠𝑀)𝑦) = ((𝑓𝑦)( ·𝑠𝑀)𝑦))
8483mpteq2dva 4903 . . . . . . . . . . . 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)))
8584oveq2d 6858 . . . . . . . . . . 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))))
86 eqid 2765 . . . . . . . . . . . . 13 (invg𝑅) = (invg𝑅)
8728ex 401 . . . . . . . . . . . . . . . . . . 19 (𝑓:𝑆𝐵 → (𝑥𝑆 → (𝑓𝑥) ∈ 𝐵))
8827, 87syl 17 . . . . . . . . . . . . . . . . . 18 (𝑓 ∈ (𝐵𝑚 𝑆) → (𝑥𝑆 → (𝑓𝑥) ∈ 𝐵))
8988com12 32 . . . . . . . . . . . . . . . . 17 (𝑥𝑆 → (𝑓 ∈ (𝐵𝑚 𝑆) → (𝑓𝑥) ∈ 𝐵))
9089ad2antll 720 . . . . . . . . . . . . . . . 16 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓 ∈ (𝐵𝑚 𝑆) → (𝑓𝑥) ∈ 𝐵))
9190com12 32 . . . . . . . . . . . . . . 15 (𝑓 ∈ (𝐵𝑚 𝑆) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
9291adantr 472 . . . . . . . . . . . . . 14 ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) → (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → (𝑓𝑥) ∈ 𝐵))
9392imp 395 . . . . . . . . . . . . 13 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (𝑓𝑥) ∈ 𝐵)
9414, 15, 17, 62, 16, 86, 26, 36, 93lmodvsneg 19176 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥))
9552eqcomi 2774 . . . . . . . . . . . . . 14 ((invg𝑅)‘(𝑓𝑥)) = 𝑌
9695a1i 11 . . . . . . . . . . . . 13 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑅)‘(𝑓𝑥)) = 𝑌)
9796oveq1d 6857 . . . . . . . . . . . 12 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → (((invg𝑅)‘(𝑓𝑥))( ·𝑠𝑀)𝑥) = (𝑌( ·𝑠𝑀)𝑥))
9894, 97eqtrd 2799 . . . . . . . . . . 11 (((𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ) ∧ ((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆))) → ((invg𝑀)‘((𝑓𝑥)( ·𝑠𝑀)𝑥)) = (𝑌( ·𝑠𝑀)𝑥))
9985, 98eqeq12d 2780 . . . . . . . . . 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)))
10099biimpd 220 . . . . . . . . 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)))
10177, 100sylbid 231 . . . . . . . 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)))
10266, 101syl5bi 233 . . . . . . 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)))
10365, 102sylbid 231 . . . . . 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)))
104103ex 401 . . . . 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))))
105104com23 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))))
1061053impia 1145 . . 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)))
107106com12 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)))
1081073impia 1145 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  Vcvv 3350  cdif 3729  wss 3732  𝒫 cpw 4315  {csn 4334   class class class wbr 4809  cmpt 4888  cres 5279  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060   finSupp cfsupp 8482  Basecbs 16130  +gcplusg 16214  Scalarcsca 16217   ·𝑠 cvsca 16218  0gc0g 16366   Σg cgsu 16367  Grpcgrp 17689  invgcminusg 17690  LModclmod 19132   linC clinc 42862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-0g 16368  df-gsum 16369  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-submnd 17602  df-grp 17692  df-minusg 17693  df-mulg 17808  df-cntz 18013  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-lmod 19134  df-linc 42864
This theorem is referenced by:  lindslinindsimp2lem5  42920
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