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Theorem frlmlbs 21738
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 2728 . . . 4 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
41, 2, 3uvcff 21732 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ π‘ˆ:𝐼⟢(Baseβ€˜πΉ))
54frnd 6735 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ran π‘ˆ βŠ† (Baseβ€˜πΉ))
6 suppssdm 8188 . . . . . 6 (π‘Ž supp (0gβ€˜π‘…)) βŠ† dom π‘Ž
7 eqid 2728 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
82, 7, 3frlmbasf 21701 . . . . . . 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 6747 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ π‘Ž ∈ (Baseβ€˜πΉ)) β†’ (π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼)
1110ralrimiva 3143 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ βˆ€π‘Ž ∈ (Baseβ€˜πΉ)(π‘Ž supp (0gβ€˜π‘…)) βŠ† 𝐼)
12 rabid2 3463 . . . 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 4004 . . . 4 𝐼 βŠ† 𝐼
15 eqid 2728 . . . . 5 (LSpanβ€˜πΉ) = (LSpanβ€˜πΉ)
16 eqid 2728 . . . . 5 (0gβ€˜π‘…) = (0gβ€˜π‘…)
17 eqid 2728 . . . . 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 21737 . . . 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 6727 . . . . 5 (π‘ˆ:𝐼⟢(Baseβ€˜πΉ) β†’ π‘ˆ Fn 𝐼)
21 fnima 6690 . . . . 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 6906 . . 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 2775 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ((LSpanβ€˜πΉ)β€˜ran π‘ˆ) = (Baseβ€˜πΉ))
25 eqid 2728 . . . . . 6 ( ·𝑠 β€˜πΉ) = ( ·𝑠 β€˜πΉ)
26 eqid 2728 . . . . . 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 4133 . . . . . 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 4670 . . . . . . 7 𝑐 ∈ {𝑐}
31 snssi 4816 . . . . . . . . 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 4261 . . . . . . . 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 2836 . . . . . 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 21694 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ 𝑅 = (Scalarβ€˜πΉ))
3736fveq2d 6906 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (Baseβ€˜π‘…) = (Baseβ€˜(Scalarβ€˜πΉ)))
3836fveq2d 6906 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (0gβ€˜π‘…) = (0gβ€˜(Scalarβ€˜πΉ)))
3938sneqd 4644 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ {(0gβ€˜π‘…)} = {(0gβ€˜(Scalarβ€˜πΉ))})
4037, 39difeq12d 4123 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)}) = ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))
4140eleq2d 2815 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (𝑏 ∈ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)}) ↔ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))})))
4241biimpar 476 . . . . . . 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 21736 . . . . 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 20467 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑏 ∈ ((Baseβ€˜π‘…) βˆ– {(0gβ€˜π‘…)})) β†’ 𝑅 ∈ NzRing)
4627, 43, 45syl2anc 582 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) ∧ (𝑐 ∈ 𝐼 ∧ 𝑏 ∈ ((Baseβ€˜(Scalarβ€˜πΉ)) βˆ– {(0gβ€˜(Scalarβ€˜πΉ))}))) β†’ 𝑅 ∈ NzRing)
471, 2, 3uvcf1 21733 . . . . . . . . . 10 ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑉) β†’ π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ))
4846, 28, 47syl2anc 582 . . . . . . . . 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 6558 . . . . . . . . . 10 (π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ) ↔ (π‘ˆ:𝐼⟢(Baseβ€˜πΉ) ∧ Fun β—‘π‘ˆ))
5049simprbi 495 . . . . . . . . 9 (π‘ˆ:𝐼–1-1β†’(Baseβ€˜πΉ) β†’ Fun β—‘π‘ˆ)
51 imadif 6642 . . . . . . . . 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 6799 . . . . . . . . . 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 6982 . . . . . . . . . . 11 ((π‘ˆ Fn 𝐼 ∧ 𝑐 ∈ 𝐼) β†’ {(π‘ˆβ€˜π‘)} = (π‘ˆ β€œ {𝑐}))
5855, 56, 57syl2anc 582 . . . . . . . . . 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 2734 . . . . . . . . 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 4123 . . . . . . . 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 2769 . . . . . . 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 6906 . . . . . 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 21737 . . . . . . 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 2768 . . . . 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 2852 . . . 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 3198 . . 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 7434 . . . . . . . 8 (π‘Ž = (π‘ˆβ€˜π‘) β†’ (𝑏( ·𝑠 β€˜πΉ)π‘Ž) = (𝑏( ·𝑠 β€˜πΉ)(π‘ˆβ€˜π‘)))
69 sneq 4642 . . . . . . . . . 10 (π‘Ž = (π‘ˆβ€˜π‘) β†’ {π‘Ž} = {(π‘ˆβ€˜π‘)})
7069difeq2d 4122 . . . . . . . . 9 (π‘Ž = (π‘ˆβ€˜π‘) β†’ (ran π‘ˆ βˆ– {π‘Ž}) = (ran π‘ˆ βˆ– {(π‘ˆβ€˜π‘)}))
7170fveq2d 6906 . . . . . . . 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 2823 . . . . . . 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 3175 . . . . 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 7103 . . . 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 7460 . . 3 𝐹 ∈ V
79 eqid 2728 . . . 4 (Scalarβ€˜πΉ) = (Scalarβ€˜πΉ)
80 eqid 2728 . . . 4 (Baseβ€˜(Scalarβ€˜πΉ)) = (Baseβ€˜(Scalarβ€˜πΉ))
81 frlmlbs.j . . . 4 𝐽 = (LBasisβ€˜πΉ)
82 eqid 2728 . . . 4 (0gβ€˜(Scalarβ€˜πΉ)) = (0gβ€˜(Scalarβ€˜πΉ))
833, 79, 25, 80, 81, 15, 82islbs 20968 . . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430  Vcvv 3473   βˆ– cdif 3946   βŠ† wss 3949  {csn 4632  β—‘ccnv 5681  ran crn 5683   β€œ cima 5685  Fun wfun 6547   Fn wfn 6548  βŸΆwf 6549  β€“1-1β†’wf1 6550  β€˜cfv 6553  (class class class)co 7426   supp csupp 8171  Basecbs 17187  Scalarcsca 17243   ·𝑠 cvsca 17244  0gc0g 17428  Ringcrg 20180  NzRingcnzr 20458  LSpanclspn 20862  LBasisclbs 20966   freeLMod cfrlm 21687   unitVec cuvc 21723
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-supp 8172  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-fsupp 9394  df-sup 9473  df-oi 9541  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-fzo 13668  df-seq 14007  df-hash 14330  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-ip 17258  df-tset 17259  df-ple 17260  df-ds 17262  df-hom 17264  df-cco 17265  df-0g 17430  df-gsum 17431  df-prds 17436  df-pws 17438  df-mre 17573  df-mrc 17574  df-acs 17576  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-mhm 18747  df-submnd 18748  df-grp 18900  df-minusg 18901  df-sbg 18902  df-mulg 19031  df-subg 19085  df-ghm 19175  df-cntz 19275  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-ur 20129  df-ring 20182  df-nzr 20459  df-subrg 20515  df-lmod 20752  df-lss 20823  df-lsp 20863  df-lmhm 20914  df-lbs 20967  df-sra 21065  df-rgmod 21066  df-dsmm 21673  df-frlm 21688  df-uvc 21724
This theorem is referenced by:  frlmup3  21741  frlmup4  21742  lmisfree  21783  frlmisfrlm  21789  frlmdim  33342  lindsdom  37120  aacllem  48312
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