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Theorem frlmlbs 19897
Description: The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
Hypotheses
Ref Expression
frlmlbs.f 𝐹 = (𝑅 freeLMod 𝐼)
frlmlbs.u 𝑈 = (𝑅 unitVec 𝐼)
frlmlbs.j 𝐽 = (LBasis‘𝐹)
Assertion
Ref Expression
frlmlbs ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ran 𝑈𝐽)

Proof of Theorem frlmlbs
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frlmlbs.u . . . 4 𝑈 = (𝑅 unitVec 𝐼)
2 frlmlbs.f . . . 4 𝐹 = (𝑅 freeLMod 𝐼)
3 eqid 2609 . . . 4 (Base‘𝐹) = (Base‘𝐹)
41, 2, 3uvcff 19891 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → 𝑈:𝐼⟶(Base‘𝐹))
5 frn 5952 . . 3 (𝑈:𝐼⟶(Base‘𝐹) → ran 𝑈 ⊆ (Base‘𝐹))
64, 5syl 17 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ran 𝑈 ⊆ (Base‘𝐹))
7 eqid 2609 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
82, 7, 3frlmbasf 19865 . . . . . . 7 ((𝐼𝑉𝑎 ∈ (Base‘𝐹)) → 𝑎:𝐼⟶(Base‘𝑅))
98adantll 745 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → 𝑎:𝐼⟶(Base‘𝑅))
10 suppssdm 7172 . . . . . . 7 (𝑎 supp (0g𝑅)) ⊆ dom 𝑎
11 fdm 5950 . . . . . . 7 (𝑎:𝐼⟶(Base‘𝑅) → dom 𝑎 = 𝐼)
1210, 11syl5sseq 3615 . . . . . 6 (𝑎:𝐼⟶(Base‘𝑅) → (𝑎 supp (0g𝑅)) ⊆ 𝐼)
139, 12syl 17 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ 𝑎 ∈ (Base‘𝐹)) → (𝑎 supp (0g𝑅)) ⊆ 𝐼)
1413ralrimiva 2948 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g𝑅)) ⊆ 𝐼)
15 rabid2 3095 . . . 4 ((Base‘𝐹) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ 𝐼} ↔ ∀𝑎 ∈ (Base‘𝐹)(𝑎 supp (0g𝑅)) ⊆ 𝐼)
1614, 15sylibr 222 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → (Base‘𝐹) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ 𝐼})
17 ssid 3586 . . . 4 𝐼𝐼
18 eqid 2609 . . . . 5 (LSpan‘𝐹) = (LSpan‘𝐹)
19 eqid 2609 . . . . 5 (0g𝑅) = (0g𝑅)
20 eqid 2609 . . . . 5 {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ 𝐼} = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ 𝐼}
212, 1, 18, 3, 19, 20frlmsslsp 19896 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑉𝐼𝐼) → ((LSpan‘𝐹)‘(𝑈𝐼)) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ 𝐼})
2217, 21mp3an3 1404 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ((LSpan‘𝐹)‘(𝑈𝐼)) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ 𝐼})
23 ffn 5944 . . . . 5 (𝑈:𝐼⟶(Base‘𝐹) → 𝑈 Fn 𝐼)
24 fnima 5909 . . . . 5 (𝑈 Fn 𝐼 → (𝑈𝐼) = ran 𝑈)
254, 23, 243syl 18 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → (𝑈𝐼) = ran 𝑈)
2625fveq2d 6092 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ((LSpan‘𝐹)‘(𝑈𝐼)) = ((LSpan‘𝐹)‘ran 𝑈))
2716, 22, 263eqtr2rd 2650 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ((LSpan‘𝐹)‘ran 𝑈) = (Base‘𝐹))
28 eqid 2609 . . . . . 6 ( ·𝑠𝐹) = ( ·𝑠𝐹)
29 eqid 2609 . . . . . 6 {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ (𝐼 ∖ {𝑐})} = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ (𝐼 ∖ {𝑐})}
30 simpll 785 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑅 ∈ Ring)
31 simplr 787 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝐼𝑉)
32 difssd 3699 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ {𝑐}) ⊆ 𝐼)
33 vsnid 4155 . . . . . . 7 𝑐 ∈ {𝑐}
34 snssi 4279 . . . . . . . . 9 (𝑐𝐼 → {𝑐} ⊆ 𝐼)
3534ad2antrl 759 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → {𝑐} ⊆ 𝐼)
36 dfss4 3819 . . . . . . . 8 ({𝑐} ⊆ 𝐼 ↔ (𝐼 ∖ (𝐼 ∖ {𝑐})) = {𝑐})
3735, 36sylib 206 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝐼 ∖ (𝐼 ∖ {𝑐})) = {𝑐})
3833, 37syl5eleqr 2694 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐 ∈ (𝐼 ∖ (𝐼 ∖ {𝑐})))
392frlmsca 19858 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → 𝑅 = (Scalar‘𝐹))
4039fveq2d 6092 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → (Base‘𝑅) = (Base‘(Scalar‘𝐹)))
4139fveq2d 6092 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → (0g𝑅) = (0g‘(Scalar‘𝐹)))
4241sneqd 4136 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → {(0g𝑅)} = {(0g‘(Scalar‘𝐹))})
4340, 42difeq12d 3690 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ((Base‘𝑅) ∖ {(0g𝑅)}) = ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))
4443eleq2d 2672 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → (𝑏 ∈ ((Base‘𝑅) ∖ {(0g𝑅)}) ↔ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))})))
4544biimpar 500 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ 𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))})) → 𝑏 ∈ ((Base‘𝑅) ∖ {(0g𝑅)}))
4645adantrl 747 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑏 ∈ ((Base‘𝑅) ∖ {(0g𝑅)}))
472, 1, 3, 7, 28, 19, 29, 30, 31, 32, 38, 46frlmssuvc2 19895 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ¬ (𝑏( ·𝑠𝐹)(𝑈𝑐)) ∈ {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ (𝐼 ∖ {𝑐})})
4819, 7ringelnzr 19033 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑏 ∈ ((Base‘𝑅) ∖ {(0g𝑅)})) → 𝑅 ∈ NzRing)
4930, 46, 48syl2anc 690 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑅 ∈ NzRing)
501, 2, 3uvcf1 19892 . . . . . . . . . 10 ((𝑅 ∈ NzRing ∧ 𝐼𝑉) → 𝑈:𝐼1-1→(Base‘𝐹))
5149, 31, 50syl2anc 690 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑈:𝐼1-1→(Base‘𝐹))
52 df-f1 5795 . . . . . . . . . 10 (𝑈:𝐼1-1→(Base‘𝐹) ↔ (𝑈:𝐼⟶(Base‘𝐹) ∧ Fun 𝑈))
5352simprbi 478 . . . . . . . . 9 (𝑈:𝐼1-1→(Base‘𝐹) → Fun 𝑈)
54 imadif 5873 . . . . . . . . 9 (Fun 𝑈 → (𝑈 “ (𝐼 ∖ {𝑐})) = ((𝑈𝐼) ∖ (𝑈 “ {𝑐})))
5551, 53, 543syl 18 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ (𝐼 ∖ {𝑐})) = ((𝑈𝐼) ∖ (𝑈 “ {𝑐})))
56 f1fn 6000 . . . . . . . . . 10 (𝑈:𝐼1-1→(Base‘𝐹) → 𝑈 Fn 𝐼)
5751, 56, 243syl 18 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈𝐼) = ran 𝑈)
5851, 56syl 17 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑈 Fn 𝐼)
59 simprl 789 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → 𝑐𝐼)
60 fnsnfv 6153 . . . . . . . . . . 11 ((𝑈 Fn 𝐼𝑐𝐼) → {(𝑈𝑐)} = (𝑈 “ {𝑐}))
6158, 59, 60syl2anc 690 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → {(𝑈𝑐)} = (𝑈 “ {𝑐}))
6261eqcomd 2615 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (𝑈 “ {𝑐}) = {(𝑈𝑐)})
6357, 62difeq12d 3690 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ((𝑈𝐼) ∖ (𝑈 “ {𝑐})) = (ran 𝑈 ∖ {(𝑈𝑐)}))
6455, 63eqtr2d 2644 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → (ran 𝑈 ∖ {(𝑈𝑐)}) = (𝑈 “ (𝐼 ∖ {𝑐})))
6564fveq2d 6092 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈𝑐)})) = ((LSpan‘𝐹)‘(𝑈 “ (𝐼 ∖ {𝑐}))))
662, 1, 18, 3, 19, 29frlmsslsp 19896 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼𝑉 ∧ (𝐼 ∖ {𝑐}) ⊆ 𝐼) → ((LSpan‘𝐹)‘(𝑈 “ (𝐼 ∖ {𝑐}))) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ (𝐼 ∖ {𝑐})})
6730, 31, 32, 66syl3anc 1317 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ((LSpan‘𝐹)‘(𝑈 “ (𝐼 ∖ {𝑐}))) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ (𝐼 ∖ {𝑐})})
6865, 67eqtrd 2643 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈𝑐)})) = {𝑎 ∈ (Base‘𝐹) ∣ (𝑎 supp (0g𝑅)) ⊆ (𝐼 ∖ {𝑐})})
6947, 68neleqtrrd 2709 . . . 4 (((𝑅 ∈ Ring ∧ 𝐼𝑉) ∧ (𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}))) → ¬ (𝑏( ·𝑠𝐹)(𝑈𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈𝑐)})))
7069ralrimivva 2953 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ∀𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠𝐹)(𝑈𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈𝑐)})))
71 oveq2 6535 . . . . . . . 8 (𝑎 = (𝑈𝑐) → (𝑏( ·𝑠𝐹)𝑎) = (𝑏( ·𝑠𝐹)(𝑈𝑐)))
72 sneq 4134 . . . . . . . . . 10 (𝑎 = (𝑈𝑐) → {𝑎} = {(𝑈𝑐)})
7372difeq2d 3689 . . . . . . . . 9 (𝑎 = (𝑈𝑐) → (ran 𝑈 ∖ {𝑎}) = (ran 𝑈 ∖ {(𝑈𝑐)}))
7473fveq2d 6092 . . . . . . . 8 (𝑎 = (𝑈𝑐) → ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) = ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈𝑐)})))
7571, 74eleq12d 2681 . . . . . . 7 (𝑎 = (𝑈𝑐) → ((𝑏( ·𝑠𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) ↔ (𝑏( ·𝑠𝐹)(𝑈𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈𝑐)}))))
7675notbid 306 . . . . . 6 (𝑎 = (𝑈𝑐) → (¬ (𝑏( ·𝑠𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) ↔ ¬ (𝑏( ·𝑠𝐹)(𝑈𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈𝑐)}))))
7776ralbidv 2968 . . . . 5 (𝑎 = (𝑈𝑐) → (∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) ↔ ∀𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠𝐹)(𝑈𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈𝑐)}))))
7877ralrn 6255 . . . 4 (𝑈 Fn 𝐼 → (∀𝑎 ∈ ran 𝑈𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) ↔ ∀𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠𝐹)(𝑈𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈𝑐)}))))
794, 23, 783syl 18 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → (∀𝑎 ∈ ran 𝑈𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})) ↔ ∀𝑐𝐼𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠𝐹)(𝑈𝑐)) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {(𝑈𝑐)}))))
8070, 79mpbird 245 . 2 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ∀𝑎 ∈ ran 𝑈𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})))
81 ovex 6555 . . . 4 (𝑅 freeLMod 𝐼) ∈ V
822, 81eqeltri 2683 . . 3 𝐹 ∈ V
83 eqid 2609 . . . 4 (Scalar‘𝐹) = (Scalar‘𝐹)
84 eqid 2609 . . . 4 (Base‘(Scalar‘𝐹)) = (Base‘(Scalar‘𝐹))
85 frlmlbs.j . . . 4 𝐽 = (LBasis‘𝐹)
86 eqid 2609 . . . 4 (0g‘(Scalar‘𝐹)) = (0g‘(Scalar‘𝐹))
873, 83, 28, 84, 85, 18, 86islbs 18843 . . 3 (𝐹 ∈ V → (ran 𝑈𝐽 ↔ (ran 𝑈 ⊆ (Base‘𝐹) ∧ ((LSpan‘𝐹)‘ran 𝑈) = (Base‘𝐹) ∧ ∀𝑎 ∈ ran 𝑈𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎})))))
8882, 87ax-mp 5 . 2 (ran 𝑈𝐽 ↔ (ran 𝑈 ⊆ (Base‘𝐹) ∧ ((LSpan‘𝐹)‘ran 𝑈) = (Base‘𝐹) ∧ ∀𝑎 ∈ ran 𝑈𝑏 ∈ ((Base‘(Scalar‘𝐹)) ∖ {(0g‘(Scalar‘𝐹))}) ¬ (𝑏( ·𝑠𝐹)𝑎) ∈ ((LSpan‘𝐹)‘(ran 𝑈 ∖ {𝑎}))))
896, 27, 80, 88syl3anbrc 1238 1 ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ran 𝑈𝐽)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  {crab 2899  Vcvv 3172  cdif 3536  wss 3539  {csn 4124  ccnv 5027  dom cdm 5028  ran crn 5029  cima 5031  Fun wfun 5784   Fn wfn 5785  wf 5786  1-1wf1 5787  cfv 5790  (class class class)co 6527   supp csupp 7159  Basecbs 15641  Scalarcsca 15717   ·𝑠 cvsca 15718  0gc0g 15869  Ringcrg 18316  LSpanclspn 18738  LBasisclbs 18841  NzRingcnzr 19024   freeLMod cfrlm 19851   unitVec cuvc 19882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-sup 8208  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-fz 12153  df-fzo 12290  df-seq 12619  df-hash 12935  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mulr 15728  df-sca 15730  df-vsca 15731  df-ip 15732  df-tset 15733  df-ple 15734  df-ds 15737  df-hom 15739  df-cco 15740  df-0g 15871  df-gsum 15872  df-prds 15877  df-pws 15879  df-mre 16015  df-mrc 16016  df-acs 16018  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-mhm 17104  df-submnd 17105  df-grp 17194  df-minusg 17195  df-sbg 17196  df-mulg 17310  df-subg 17360  df-ghm 17427  df-cntz 17519  df-cmn 17964  df-abl 17965  df-mgp 18259  df-ur 18271  df-ring 18318  df-subrg 18547  df-lmod 18634  df-lss 18700  df-lsp 18739  df-lmhm 18789  df-lbs 18842  df-sra 18939  df-rgmod 18940  df-nzr 19025  df-dsmm 19837  df-frlm 19852  df-uvc 19883
This theorem is referenced by:  frlmup3  19900  frlmup4  19901  lmisfree  19942  frlmisfrlm  19948  lindsdom  32376  aacllem  42319
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