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Theorem lindslinindimp2lem2 42758
Description: Lemma 2 for lindslinindsimp2 42762. (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 8045 . . . . . 6 (𝑓 ∈ (𝐵𝑚 𝑆) → 𝑓:𝑆𝐵)
213ad2ant3 1130 . . . . 5 ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆)) → 𝑓:𝑆𝐵)
32adantl 473 . . . 4 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝑓:𝑆𝐵)
4 difss 3880 . . . 4 (𝑆 ∖ {𝑥}) ⊆ 𝑆
5 fssres 6231 . . . 4 ((𝑓:𝑆𝐵 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑆) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
63, 4, 5sylancl 697 . . 3 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
7 lindslinind.g . . . 4 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))
87feq1i 6197 . . 3 (𝐺:(𝑆 ∖ {𝑥})⟶𝐵 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
96, 8sylibr 224 . 2 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝐺:(𝑆 ∖ {𝑥})⟶𝐵)
10 lindslinind.b . . . 4 𝐵 = (Base‘𝑅)
11 fvex 6362 . . . 4 (Base‘𝑅) ∈ V
1210, 11eqeltri 2835 . . 3 𝐵 ∈ V
13 difexg 4960 . . . 4 (𝑆𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
1413ad2antrr 764 . . 3 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → (𝑆 ∖ {𝑥}) ∈ V)
15 elmapg 8036 . . 3 ((𝐵 ∈ V ∧ (𝑆 ∖ {𝑥}) ∈ V) → (𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵))
1612, 14, 15sylancr 698 . 2 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → (𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵))
179, 16mpbird 247 1 (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  cdif 3712  wss 3715  {csn 4321  cres 5268  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  Basecbs 16059  Scalarcsca 16146  0gc0g 16302  invgcminusg 17624  LModclmod 19065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025
This theorem is referenced by:  lindslinindimp2lem4  42760  lindslinindsimp2lem5  42761
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