Theorem lindslinindimp2lem2 42042

Description: Lemma 2 for lindslinindsimp2 42046. (Contributed by AV, 25-Apr-2019.)

Hypotheses

Ref	Expression
lindslinind.r	⊢ 𝑅 = (Scalar‘𝑀)
lindslinind.b	⊢ 𝐵 = (Base‘𝑅)
lindslinind.0	⊢ 0 = (0g‘𝑅)
lindslinind.z	⊢ 𝑍 = (0g‘𝑀)
lindslinind.y	⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥))
lindslinind.g	⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))

Assertion

Ref	Expression
lindslinindimp2lem2	⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑𝑚 𝑆))) → 𝐺 ∈ (𝐵 ↑𝑚 (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

Step	Hyp	Ref	Expression
1		elmapi 7765	. . . . . 6 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝑆) → 𝑓:𝑆⟶𝐵)
2	1	3ad2ant3 1077	. . . . 5 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑𝑚 𝑆)) → 𝑓:𝑆⟶𝐵)
3	2	adantl 481	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑𝑚 𝑆))) → 𝑓:𝑆⟶𝐵)
4		difss 3699	. . . 4 ⊢ (𝑆 ∖ {𝑥}) ⊆ 𝑆
5		fssres 5983	. . . 4 ⊢ ((𝑓:𝑆⟶𝐵 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑆) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
6	3, 4, 5	sylancl 693	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑𝑚 𝑆))) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
7		lindslinind.g	. . . 4 ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))
8	7	feq1i 5949	. . 3 ⊢ (𝐺:(𝑆 ∖ {𝑥})⟶𝐵 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
9	6, 8	sylibr 223	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑𝑚 𝑆))) → 𝐺:(𝑆 ∖ {𝑥})⟶𝐵)
10		lindslinind.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
11		fvex 6113	. . . 4 ⊢ (Base‘𝑅) ∈ V
12	10, 11	eqeltri 2684	. . 3 ⊢ 𝐵 ∈ V
13		difexg 4735	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
14	13	ad2antrr 758	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑𝑚 𝑆))) → (𝑆 ∖ {𝑥}) ∈ V)
15		elmapg 7757	. . 3 ⊢ ((𝐵 ∈ V ∧ (𝑆 ∖ {𝑥}) ∈ V) → (𝐺 ∈ (𝐵 ↑𝑚 (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵))
16	12, 14, 15	sylancr 694	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑𝑚 𝑆))) → (𝐺 ∈ (𝐵 ↑𝑚 (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵))
17	9, 16	mpbird 246	1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑𝑚 𝑆))) → 𝐺 ∈ (𝐵 ↑𝑚 (𝑆 ∖ {𝑥})))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  Basecbs 15695  Scalarcsca 15771  0_gc0g 15923  inv_gcminusg 17246  LModclmod 18686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746

This theorem is referenced by:  lindslinindimp2lem4  42044  lindslinindsimp2lem5  42045