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Theorem lindslinindimp2lem2 44014
Description: Lemma 2 for lindslinindsimp2 44018. (Contributed by AV, 25-Apr-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
lindslinindimp2lem2 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})))
Distinct variable groups:   𝐵,𝑓   𝑓,𝑀   𝑅,𝑓,𝑥   𝑆,𝑓,𝑥   𝑓,𝑍   0 ,𝑓,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐺(𝑥,𝑓)   𝑀(𝑥)   𝑉(𝑥,𝑓)   𝑌(𝑥,𝑓)   𝑍(𝑥)

Proof of Theorem lindslinindimp2lem2
StepHypRef Expression
1 elmapi 8278 . . . . . 6 (𝑓 ∈ (𝐵𝑚 𝑆) → 𝑓:𝑆𝐵)
213ad2ant3 1128 . . . . 5 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆)) → 𝑓:𝑆𝐵)
32adantl 482 . . . 4 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝑓:𝑆𝐵)
4 difss 4029 . . . 4 (𝑆 ∖ {𝑥}) ⊆ 𝑆
5 fssres 6412 . . . 4 ((𝑓:𝑆𝐵 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑆) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
63, 4, 5sylancl 586 . . 3 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
7 lindslinind.g . . . 4 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))
87feq1i 6373 . . 3 (𝐺:(𝑆 ∖ {𝑥})⟶𝐵 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
96, 8sylibr 235 . 2 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝐺:(𝑆 ∖ {𝑥})⟶𝐵)
10 lindslinind.b . . . 4 𝐵 = (Base‘𝑅)
1110fvexi 6552 . . 3 𝐵 ∈ V
12 difexg 5122 . . . 4 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
1312ad2antrr 722 . . 3 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → (𝑆 ∖ {𝑥}) ∈ V)
14 elmapg 8269 . . 3 ((𝐵 ∈ V ∧ (𝑆 ∖ {𝑥}) ∈ V) → (𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵))
1511, 13, 14sylancr 587 . 2 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → (𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵))
169, 15mpbird 258 1 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  Vcvv 3437  cdif 3856  wss 3859  {csn 4472  cres 5445  wf 6221  cfv 6225  (class class class)co 7016  𝑚 cmap 8256  Basecbs 16312  Scalarcsca 16397  0gc0g 16542  invgcminusg 17862  LModclmod 19324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-map 8258
This theorem is referenced by:  lindslinindimp2lem4  44016  lindslinindsimp2lem5  44017
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