Theorem lindslinindimp2lem2 48947

Description: Lemma 2 for lindslinindsimp2 48951. (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 8789	. . . . . 6 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → 𝑓:𝑆⟶𝐵)
2	1	3ad2ant3 1136	. . . . 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 4077	. . . 4 ⊢ (𝑆 ∖ {𝑥}) ⊆ 𝑆
5		fssres 6700	. . . 4 ⊢ ((𝑓:𝑆⟶𝐵 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑆) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
6	3, 4, 5	sylancl 587	. . 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 6653	. . 3 ⊢ (𝐺:(𝑆 ∖ {𝑥})⟶𝐵 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵)
9	6, 8	sylibr 234	. 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 6848	. . 3 ⊢ 𝐵 ∈ V
12		difexg 5266	. . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V)
13	12	ad2antrr 727	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → (𝑆 ∖ {𝑥}) ∈ V)
14		elmapg 8779	. . 3 ⊢ ((𝐵 ∈ V ∧ (𝑆 ∖ {𝑥}) ∈ V) → (𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵))
15	11, 13, 14	sylancr 588	. 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 257	1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  {csn 4568   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ↑_m cmap 8766  Basecbs 17170  Scalarcsca 17214  0_gc0g 17393  inv_gcminusg 18901  LModclmod 20846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8768

This theorem is referenced by:  lindslinindimp2lem4  48949  lindslinindsimp2lem5  48950