Theorem lindslinindimp2lem2 49090

Description: Lemma 2 for lindslinindsimp2 49094. (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 8834	. . . . . 6 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → 𝑓:𝑆⟶𝐵)
2	1	3ad2ant3 1151	. . . . 5 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆)) → 𝑓:𝑆⟶𝐵)
3	2	adantl 486	. . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝑓:𝑆⟶𝐵)
4		difss 4092	. . . 4 ⊢ (𝑆 ∖ {𝑥}) ⊆ 𝑆
5		fssres 6734	. . . 4 ⊢ ((𝑓:𝑆⟶𝐵 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑆) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
6	3, 4, 5	sylancl 597	. . 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 6686	. . 3 ⊢ (𝐺:(𝑆 ∖ {𝑥})⟶𝐵 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
9	6, 8	sylibr 237	. 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	10	fvexi 6885	. . 3 ⊢ 𝐵 ∈ V
12		difexg 5290	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
13	12	ad2antrr 738	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → (𝑆 ∖ {𝑥}) ∈ V)
14		elmapg 8824	. . 3 ⊢ ((𝐵 ∈ V ∧ (𝑆 ∖ {𝑥}) ∈ V) → (𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵))
15	11, 13, 14	sylancr 598	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → (𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵))
16	9, 15	mpbird 260	1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∖ cdif 3904   ⊆ wss 3907  {csn 4585   ↾ cres 5654  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ↑_m cmap 8812  Basecbs 17259  Scalarcsca 17303  0_gc0g 17482  inv_gcminusg 18991  LModclmod 20950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814

This theorem is referenced by:  lindslinindimp2lem4  49092  lindslinindsimp2lem5  49093