MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frlmlbs Structured version   Visualization version   GIF version
Theorem frlmlbs 21687
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 π‘ˆ = (𝑅 unitVec 𝐼)
frlmlbs.j 𝐽 = (LBasisβ€˜πΉ)
Assertion
Ref Expression
frlmlbs ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ran π‘ˆ ∈ 𝐽)
Proof of Theorem frlmlbs
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frlmlbs.u . . . 4 π‘ˆ = (𝑅 unitVec 𝐼)
2 frlmlbs.f . . . 4 𝐹 = (𝑅 freeLMod 𝐼)
3 eqid 2726 . . . 4 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
41, 2, 3uvcff 21681 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ π‘ˆ:𝐼⟢(Baseβ€˜πΉ))
54frnd 6718 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ran π‘ˆ βŠ† (Baseβ€˜πΉ))
6 suppssdm 8159 . . . . . 6 (π‘Ž supp (0gβ€˜π‘…)) βŠ† dom π‘Ž
7 eqid 2726 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
82, 7, 3frlmbasf 21650 . . . . . . 7 ((𝐼 ∈ 𝑉 ∧ π‘Ž ∈ (Baseβ€˜πΉ)) β†’ π‘Ž:𝐼⟢(Baseβ€˜π‘…))
98adantll 711 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ π‘Ž ∈ (Baseβ€˜πΉ)) β†’ π‘Ž:𝐼⟢(Baseβ€˜π‘…))
106, 9fssdm 6730 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ π‘Ž ∈ (Baseβ€˜πΉ)) β†’ (π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼)
1110ralrimiva 3140 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ βˆ€π‘Ž ∈ (Baseβ€˜πΉ)(π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼)
12 rabid2 3458 . . . 4 ((Baseβ€˜πΉ) = {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼} ↔ βˆ€π‘Ž ∈ (Baseβ€˜πΉ)(π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼)
1311, 12sylibr 233 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (Baseβ€˜πΉ) = {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼})
14 ssid 3999 . . . 4 𝐼 βŠ† 𝐼
15 eqid 2726 . . . . 5 (LSpanβ€˜πΉ) = (LSpanβ€˜πΉ)
16 eqid 2726 . . . . 5 (0gβ€˜π‘…) = (0gβ€˜π‘…)
17 eqid 2726 . . . . 5 {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼} = {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼}
182, 1, 15, 3, 16, 17frlmsslsp 21686 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐼 βŠ† 𝐼) β†’ ((LSpanβ€˜πΉ)β€˜(π‘ˆ β€œ 𝐼)) = {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼})
1914, 18mp3an3 1446 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ((LSpanβ€˜πΉ)β€˜(π‘ˆ β€œ 𝐼)) = {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼})
20 ffn 6710 . . . . 5 (π‘ˆ:𝐼⟢(Baseβ€˜πΉ) β†’ π‘ˆ Fn 𝐼)
21 fnima 6673 . . . . 5 (π‘ˆ Fn 𝐼 β†’ (π‘ˆ β€œ 𝐼) = ran π‘ˆ)
224, 20, 213syl 18 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (π‘ˆ β€œ 𝐼) = ran π‘ˆ)
2322fveq2d 6888 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ((LSpanβ€˜πΉ)β€˜(π‘ˆ β€œ 𝐼)) = ((LSpanβ€˜πΉ)β€˜ran π‘ˆ))
2413, 19, 233eqtr2rd 2773 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ((LSpanβ€˜πΉ)β€˜ran π‘ˆ) = (Baseβ€˜πΉ))
25 eqid 2726 . . . . . 6 ( ·𝑠 β€˜πΉ) = ( ·𝑠 β€˜πΉ)
26 eqid 2726 . . . . . 6 {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† (𝐼 βˆ– {𝑐})} = {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† (𝐼 βˆ– {𝑐})}
27 simpll 764 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝑅 ∈ Ring)
28 simplr 766 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝐼 ∈ 𝑉)
29 difssd 4127 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ (𝐼 βˆ– {𝑐}) βŠ† 𝐼)
30 vsnid 4660 . . . . . . 7 𝑐 ∈ {𝑐}
31 snssi 4806 . . . . . . . . 9 (𝑐 ∈ 𝐼 β†’ {𝑐} βŠ† 𝐼)
3231ad2antrl 725 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ {𝑐} βŠ† 𝐼)
33 dfss4 4253 . . . . . . . 8 ({𝑐} βŠ† 𝐼 ↔ (𝐼 βˆ– (𝐼 βˆ– {𝑐})) = {𝑐})
3432, 33sylib 217 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ (𝐼 βˆ– (𝐼 βˆ– {𝑐})) = {𝑐})
3530, 34eleqtrrid 2834 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝑐 ∈ (𝐼 βˆ– (𝐼 βˆ– {𝑐})))
362frlmsca 21643 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ 𝑅 = (Scalarβ€˜πΉ))
3736fveq2d 6888 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (Baseβ€˜π‘…) = (Baseβ€˜(Scalarβ€˜πΉ)))
3836fveq2d 6888 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (0gβ€˜π‘…) = (0gβ€˜(Scalarβ€˜πΉ)))
3938sneqd 4635 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ {(0gβ€˜π‘…)} = {(0gβ€˜(Scalarβ€˜πΉ))})
4037, 39difeq12d 4118 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)}) = ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))
4140eleq2d 2813 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (𝑏 ∈ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)}) ↔ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))})))
4241biimpar 477 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))})) β†’ 𝑏 ∈ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)}))
4342adantrl 713 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝑏 ∈ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)}))
442, 1, 3, 7, 25, 16, 26, 27, 28, 29, 35, 43frlmssuvc2 21685 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ Β¬ (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)) ∈ {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† (𝐼 βˆ– {𝑐})})
4516, 7ringelnzr 20420 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑏 ∈ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)})) β†’ 𝑅 ∈ NzRing)
4627, 43, 45syl2anc 583 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝑅 ∈ NzRing)
471, 2, 3uvcf1 21682 . . . . . . . . . 10 ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑉) β†’ π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ))
4846, 28, 47syl2anc 583 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ))
49 df-f1 6541 . . . . . . . . . 10 (π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ) ↔ (π‘ˆ:𝐼⟢(Baseβ€˜πΉ) ∧ Fun β—‘π‘ˆ))
5049simprbi 496 . . . . . . . . 9 (π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ) β†’ Fun β—‘π‘ˆ)
51 imadif 6625 . . . . . . . . 9 (Fun β—‘π‘ˆ β†’ (π‘ˆ β€œ (𝐼 βˆ– {𝑐})) = ((π‘ˆ β€œ 𝐼) βˆ– (π‘ˆ β€œ {𝑐})))
5248, 50, 513syl 18 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ (π‘ˆ β€œ (𝐼 βˆ– {𝑐})) = ((π‘ˆ β€œ 𝐼) βˆ– (π‘ˆ β€œ {𝑐})))
53 f1fn 6781 . . . . . . . . . 10 (π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ) β†’ π‘ˆ Fn 𝐼)
5448, 53, 213syl 18 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ (π‘ˆ β€œ 𝐼) = ran π‘ˆ)
5548, 53syl 17 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ π‘ˆ Fn 𝐼)
56 simprl 768 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝑐 ∈ 𝐼)
57 fnsnfv 6963 . . . . . . . . . . 11 ((π‘ˆ Fn 𝐼 ∧ 𝑐 ∈ 𝐼) β†’ {(π‘ˆβ€˜π‘)} = (π‘ˆ β€œ {𝑐}))
5855, 56, 57syl2anc 583 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ {(π‘ˆβ€˜π‘)} = (π‘ˆ β€œ {𝑐}))
5958eqcomd 2732 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ (π‘ˆ β€œ {𝑐}) = {(π‘ˆβ€˜π‘)})
6054, 59difeq12d 4118 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ ((π‘ˆ β€œ 𝐼) βˆ– (π‘ˆ β€œ {𝑐})) = (ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)}))
6152, 60eqtr2d 2767 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ (ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)}) = (π‘ˆ β€œ (𝐼 βˆ– {𝑐})))
6261fveq2d 6888 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)})) = ((LSpanβ€˜πΉ)β€˜(π‘ˆ β€œ (𝐼 βˆ– {𝑐}))))
632, 1, 15, 3, 16, 26frlmsslsp 21686 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ (𝐼 βˆ– {𝑐}) βŠ† 𝐼) β†’ ((LSpanβ€˜πΉ)β€˜(π‘ˆ β€œ (𝐼 βˆ– {𝑐}))) = {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† (𝐼 βˆ– {𝑐})})
6427, 28, 29, 63syl3anc 1368 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ ((LSpanβ€˜πΉ)β€˜(π‘ˆ β€œ (𝐼 βˆ– {𝑐}))) = {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† (𝐼 βˆ– {𝑐})})
6562, 64eqtrd 2766 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)})) = {π‘Ž ∈ (Baseβ€˜πΉ) ∣ (π‘Ž supp (0gβ€˜π‘…)) βŠ† (𝐼 βˆ– {𝑐})})
6644, 65neleqtrrd 2850 . . . 4 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ Β¬ (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)})))
6766ralrimivva 3194 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ βˆ€π‘ ∈ 𝐼 βˆ€π‘ ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}) Β¬ (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)})))
68 oveq2 7412 . . . . . . . 8 (π‘Ž = (π‘ˆβ€˜π‘) β†’ (𝑏( ·𝑠 β€˜πΉ)π‘Ž) = (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)))
69 sneq 4633 . . . . . . . . . 10 (π‘Ž = (π‘ˆβ€˜π‘) β†’ {π‘Ž} = {(π‘ˆβ€˜π‘)})
7069difeq2d 4117 . . . . . . . . 9 (π‘Ž = (π‘ˆβ€˜π‘) β†’ (ran π‘ˆ βˆ– {π‘Ž}) = (ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)}))
7170fveq2d 6888 . . . . . . . 8 (π‘Ž = (π‘ˆβ€˜π‘) β†’ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {π‘Ž})) = ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)})))
7268, 71eleq12d 2821 . . . . . . 7 (π‘Ž = (π‘ˆβ€˜π‘) β†’ ((𝑏( ·𝑠 β€˜πΉ)π‘Ž) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {π‘Ž})) ↔ (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)}))))
7372notbid 318 . . . . . 6 (π‘Ž = (π‘ˆβ€˜π‘) β†’ (Β¬ (𝑏( ·𝑠 β€˜πΉ)π‘Ž) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {π‘Ž})) ↔ Β¬ (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)}))))
7473ralbidv 3171 . . . . 5 (π‘Ž = (π‘ˆβ€˜π‘) β†’ (βˆ€π‘ ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}) Β¬ (𝑏( ·𝑠 β€˜πΉ)π‘Ž) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {π‘Ž})) ↔ βˆ€π‘ ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}) Β¬ (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)}))))
7574ralrn 7082 . . . 4 (π‘ˆ Fn 𝐼 β†’ (βˆ€π‘Ž ∈ ran π‘ˆβˆ€π‘ ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}) Β¬ (𝑏( ·𝑠 β€˜πΉ)π‘Ž) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {π‘Ž})) ↔ βˆ€π‘ ∈ 𝐼 βˆ€π‘ ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}) Β¬ (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)}))))
764, 20, 753syl 18 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (βˆ€π‘Ž ∈ ran π‘ˆβˆ€π‘ ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}) Β¬ (𝑏( ·𝑠 β€˜πΉ)π‘Ž) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {π‘Ž})) ↔ βˆ€π‘ ∈ 𝐼 βˆ€π‘ ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}) Β¬ (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)}))))
7767, 76mpbird 257 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ βˆ€π‘Ž ∈ ran π‘ˆβˆ€π‘ ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}) Β¬ (𝑏( ·𝑠 β€˜πΉ)π‘Ž) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {π‘Ž})))
782ovexi 7438 . . 3 𝐹 ∈ V
79 eqid 2726 . . . 4 (Scalarβ€˜πΉ) = (Scalarβ€˜πΉ)
80 eqid 2726 . . . 4 (Baseβ€˜(Scalarβ€˜πΉ)) = (Baseβ€˜(Scalarβ€˜πΉ))
81 frlmlbs.j . . . 4 𝐽 = (LBasisβ€˜πΉ)
82 eqid 2726 . . . 4 (0gβ€˜(Scalarβ€˜πΉ)) = (0gβ€˜(Scalarβ€˜πΉ))
833, 79, 25, 80, 81, 15, 82islbs 20921 . . 3 (𝐹 ∈ V β†’ (ran π‘ˆ ∈ 𝐽 ↔ (ran π‘ˆ βŠ† (Baseβ€˜πΉ) ∧ ((LSpanβ€˜πΉ)β€˜ran π‘ˆ) = (Baseβ€˜πΉ) ∧ βˆ€π‘Ž ∈ ran π‘ˆβˆ€π‘ ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}) Β¬ (𝑏( ·𝑠 β€˜πΉ)π‘Ž) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {π‘Ž})))))
8478, 83ax-mp 5 . 2 (ran π‘ˆ ∈ 𝐽 ↔ (ran π‘ˆ βŠ† (Baseβ€˜πΉ) ∧ ((LSpanβ€˜πΉ)β€˜ran π‘ˆ) = (Baseβ€˜πΉ) ∧ βˆ€π‘Ž ∈ ran π‘ˆβˆ€π‘ ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}) Β¬ (𝑏( ·𝑠 β€˜πΉ)π‘Ž) ∈ ((LSpanβ€˜πΉ)β€˜(ran π‘ˆ βˆ– {π‘Ž}))))
855, 24, 77, 84syl3anbrc 1340 1 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ran π‘ˆ ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426  Vcvv 3468   βˆ– cdif 3940   βŠ† wss 3943  {csn 4623  β—‘ccnv 5668  ran crn 5670   β€œ cima 5672  Fun wfun 6530   Fn wfn 6531  βŸΆwf 6532  β€“1-1β†’wf1 6533  β€˜cfv 6536  (class class class)co 7404   supp csupp 8143  Basecbs 17150  Scalarcsca 17206   ·𝑠 cvsca 17207  0gc0g 17391  Ringcrg 20135  NzRingcnzr 20411  LSpanclspn 20815  LBasisclbs 20919   freeLMod cfrlm 21636   unitVec cuvc 21672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8144  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fsupp 9361  df-sup 9436  df-oi 9504  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-z 12560  df-dec 12679  df-uz 12824  df-fz 13488  df-fzo 13631  df-seq 13970  df-hash 14293  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-mulr 17217  df-sca 17219  df-vsca 17220  df-ip 17221  df-tset 17222  df-ple 17223  df-ds 17225  df-hom 17227  df-cco 17228  df-0g 17393  df-gsum 17394  df-prds 17399  df-pws 17401  df-mre 17536  df-mrc 17537  df-acs 17539  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-mhm 18710  df-submnd 18711  df-grp 18863  df-minusg 18864  df-sbg 18865  df-mulg 18993  df-subg 19047  df-ghm 19136  df-cntz 19230  df-cmn 19699  df-abl 19700  df-mgp 20037  df-rng 20055  df-ur 20084  df-ring 20137  df-nzr 20412  df-subrg 20468  df-lmod 20705  df-lss 20776  df-lsp 20816  df-lmhm 20867  df-lbs 20920  df-sra 21018  df-rgmod 21019  df-dsmm 21622  df-frlm 21637  df-uvc 21673
This theorem is referenced by:  frlmup3  21690  frlmup4  21691  lmisfree  21732  frlmisfrlm  21738  frlmdim  33213  lindsdom  36994  aacllem  48104
  Copyright terms: Public domain W3C validator