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Theorem frlmlbs 21343
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 2732 . . . 4 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
41, 2, 3uvcff 21337 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ π‘ˆ:𝐼⟢(Baseβ€˜πΉ))
54frnd 6722 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ran π‘ˆ βŠ† (Baseβ€˜πΉ))
6 suppssdm 8158 . . . . . 6 (π‘Ž supp (0gβ€˜π‘…)) βŠ† dom π‘Ž
7 eqid 2732 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
82, 7, 3frlmbasf 21306 . . . . . . 7 ((𝐼 ∈ 𝑉 ∧ π‘Ž ∈ (Baseβ€˜πΉ)) β†’ π‘Ž:𝐼⟢(Baseβ€˜π‘…))
98adantll 712 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ π‘Ž ∈ (Baseβ€˜πΉ)) β†’ π‘Ž:𝐼⟢(Baseβ€˜π‘…))
106, 9fssdm 6734 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ π‘Ž ∈ (Baseβ€˜πΉ)) β†’ (π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼)
1110ralrimiva 3146 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ βˆ€π‘Ž ∈ (Baseβ€˜πΉ)(π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼)
12 rabid2 3464 . . . 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 4003 . . . 4 𝐼 βŠ† 𝐼
15 eqid 2732 . . . . 5 (LSpanβ€˜πΉ) = (LSpanβ€˜πΉ)
16 eqid 2732 . . . . 5 (0gβ€˜π‘…) = (0gβ€˜π‘…)
17 eqid 2732 . . . . 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 21342 . . . 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 1450 . . 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 6714 . . . . 5 (π‘ˆ:𝐼⟢(Baseβ€˜πΉ) β†’ π‘ˆ Fn 𝐼)
21 fnima 6677 . . . . 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 6892 . . 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 2779 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ((LSpanβ€˜πΉ)β€˜ran π‘ˆ) = (Baseβ€˜πΉ))
25 eqid 2732 . . . . . 6 ( ·𝑠 β€˜πΉ) = ( ·𝑠 β€˜πΉ)
26 eqid 2732 . . . . . 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 765 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝑅 ∈ Ring)
28 simplr 767 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝐼 ∈ 𝑉)
29 difssd 4131 . . . . . 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 4664 . . . . . . 7 𝑐 ∈ {𝑐}
31 snssi 4810 . . . . . . . . 9 (𝑐 ∈ 𝐼 β†’ {𝑐} βŠ† 𝐼)
3231ad2antrl 726 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ {𝑐} βŠ† 𝐼)
33 dfss4 4257 . . . . . . . 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 2840 . . . . . 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 21299 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ 𝑅 = (Scalarβ€˜πΉ))
3736fveq2d 6892 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (Baseβ€˜π‘…) = (Baseβ€˜(Scalarβ€˜πΉ)))
3836fveq2d 6892 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (0gβ€˜π‘…) = (0gβ€˜(Scalarβ€˜πΉ)))
3938sneqd 4639 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ {(0gβ€˜π‘…)} = {(0gβ€˜(Scalarβ€˜πΉ))})
4037, 39difeq12d 4122 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)}) = ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))
4140eleq2d 2819 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (𝑏 ∈ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)}) ↔ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))})))
4241biimpar 478 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))})) β†’ 𝑏 ∈ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)}))
4342adantrl 714 . . . . . 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 21341 . . . . 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 20292 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑏 ∈ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)})) β†’ 𝑅 ∈ NzRing)
4627, 43, 45syl2anc 584 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝑅 ∈ NzRing)
471, 2, 3uvcf1 21338 . . . . . . . . . 10 ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑉) β†’ π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ))
4846, 28, 47syl2anc 584 . . . . . . . . 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 6545 . . . . . . . . . 10 (π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ) ↔ (π‘ˆ:𝐼⟢(Baseβ€˜πΉ) ∧ Fun β—‘π‘ˆ))
5049simprbi 497 . . . . . . . . 9 (π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ) β†’ Fun β—‘π‘ˆ)
51 imadif 6629 . . . . . . . . 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 6785 . . . . . . . . . 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 769 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝑐 ∈ 𝐼)
57 fnsnfv 6967 . . . . . . . . . . 11 ((π‘ˆ Fn 𝐼 ∧ 𝑐 ∈ 𝐼) β†’ {(π‘ˆβ€˜π‘)} = (π‘ˆ β€œ {𝑐}))
5855, 56, 57syl2anc 584 . . . . . . . . . 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 2738 . . . . . . . . 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 4122 . . . . . . . 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 2773 . . . . . . 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 6892 . . . . . 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 21342 . . . . . . 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 1371 . . . . . 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 2772 . . . . 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 2856 . . . 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 3200 . . 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 7413 . . . . . . . 8 (π‘Ž = (π‘ˆβ€˜π‘) β†’ (𝑏( ·𝑠 β€˜πΉ)π‘Ž) = (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)))
69 sneq 4637 . . . . . . . . . 10 (π‘Ž = (π‘ˆβ€˜π‘) β†’ {π‘Ž} = {(π‘ˆβ€˜π‘)})
7069difeq2d 4121 . . . . . . . . 9 (π‘Ž = (π‘ˆβ€˜π‘) β†’ (ran π‘ˆ βˆ– {π‘Ž}) = (ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)}))
7170fveq2d 6892 . . . . . . . 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 2827 . . . . . . 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 317 . . . . . 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 3177 . . . . 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 7086 . . . 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 256 . 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 7439 . . 3 𝐹 ∈ V
79 eqid 2732 . . . 4 (Scalarβ€˜πΉ) = (Scalarβ€˜πΉ)
80 eqid 2732 . . . 4 (Baseβ€˜(Scalarβ€˜πΉ)) = (Baseβ€˜(Scalarβ€˜πΉ))
81 frlmlbs.j . . . 4 𝐽 = (LBasisβ€˜πΉ)
82 eqid 2732 . . . 4 (0gβ€˜(Scalarβ€˜πΉ)) = (0gβ€˜(Scalarβ€˜πΉ))
833, 79, 25, 80, 81, 15, 82islbs 20679 . . 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 1343 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432  Vcvv 3474   βˆ– cdif 3944   βŠ† wss 3947  {csn 4627  β—‘ccnv 5674  ran crn 5676   β€œ cima 5678  Fun wfun 6534   Fn wfn 6535  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€˜cfv 6540  (class class class)co 7405   supp csupp 8142  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  Ringcrg 20049  NzRingcnzr 20283  LSpanclspn 20574  LBasisclbs 20677   freeLMod cfrlm 21292   unitVec cuvc 21328
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-hom 17217  df-cco 17218  df-0g 17383  df-gsum 17384  df-prds 17389  df-pws 17391  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mulg 18945  df-subg 18997  df-ghm 19084  df-cntz 19175  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-nzr 20284  df-subrg 20353  df-lmod 20465  df-lss 20535  df-lsp 20575  df-lmhm 20625  df-lbs 20678  df-sra 20777  df-rgmod 20778  df-dsmm 21278  df-frlm 21293  df-uvc 21329
This theorem is referenced by:  frlmup3  21346  frlmup4  21347  lmisfree  21388  frlmisfrlm  21394  frlmdim  32684  lindsdom  36470  aacllem  47801
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