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Theorem frlmbas 20018
Description: Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)
Hypotheses
Ref Expression
frlmval.f 𝐹 = (𝑅 freeLMod 𝐼)
frlmbas.n 𝑁 = (Base‘𝑅)
frlmbas.z 0 = (0g𝑅)
frlmbas.b 𝐵 = {𝑘 ∈ (𝑁𝑚 𝐼) ∣ 𝑘 finSupp 0 }
Assertion
Ref Expression
frlmbas ((𝑅𝑉𝐼𝑊) → 𝐵 = (Base‘𝐹))
Distinct variable groups:   𝑘,𝑁   𝑅,𝑘   𝑘,𝐼   𝑘,𝑊   𝑘,𝑉   0 ,𝑘
Allowed substitution hints:   𝐵(𝑘)   𝐹(𝑘)

Proof of Theorem frlmbas
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvex 6158 . . . . 5 (ringLMod‘𝑅) ∈ V
2 fnconstg 6050 . . . . 5 ((ringLMod‘𝑅) ∈ V → (𝐼 × {(ringLMod‘𝑅)}) Fn 𝐼)
31, 2ax-mp 5 . . . 4 (𝐼 × {(ringLMod‘𝑅)}) Fn 𝐼
4 eqid 2621 . . . . 5 (𝑅Xs(𝐼 × {(ringLMod‘𝑅)})) = (𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))
5 eqid 2621 . . . . 5 {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin}
64, 5dsmmbas2 20000 . . . 4 (((𝐼 × {(ringLMod‘𝑅)}) Fn 𝐼𝐼𝑊) → {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = (Base‘(𝑅m (𝐼 × {(ringLMod‘𝑅)}))))
73, 6mpan 705 . . 3 (𝐼𝑊 → {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = (Base‘(𝑅m (𝐼 × {(ringLMod‘𝑅)}))))
87adantl 482 . 2 ((𝑅𝑉𝐼𝑊) → {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = (Base‘(𝑅m (𝐼 × {(ringLMod‘𝑅)}))))
9 frlmbas.b . . 3 𝐵 = {𝑘 ∈ (𝑁𝑚 𝐼) ∣ 𝑘 finSupp 0 }
10 fvco2 6230 . . . . . . . . . . . . 13 (((𝐼 × {(ringLMod‘𝑅)}) Fn 𝐼𝑥𝐼) → ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥) = (0g‘((𝐼 × {(ringLMod‘𝑅)})‘𝑥)))
113, 10mpan 705 . . . . . . . . . . . 12 (𝑥𝐼 → ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥) = (0g‘((𝐼 × {(ringLMod‘𝑅)})‘𝑥)))
1211adantl 482 . . . . . . . . . . 11 ((((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) ∧ 𝑥𝐼) → ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥) = (0g‘((𝐼 × {(ringLMod‘𝑅)})‘𝑥)))
131fvconst2 6423 . . . . . . . . . . . . . 14 (𝑥𝐼 → ((𝐼 × {(ringLMod‘𝑅)})‘𝑥) = (ringLMod‘𝑅))
1413adantl 482 . . . . . . . . . . . . 13 ((((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) ∧ 𝑥𝐼) → ((𝐼 × {(ringLMod‘𝑅)})‘𝑥) = (ringLMod‘𝑅))
1514fveq2d 6152 . . . . . . . . . . . 12 ((((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) ∧ 𝑥𝐼) → (0g‘((𝐼 × {(ringLMod‘𝑅)})‘𝑥)) = (0g‘(ringLMod‘𝑅)))
16 frlmbas.z . . . . . . . . . . . . 13 0 = (0g𝑅)
17 rlm0 19116 . . . . . . . . . . . . 13 (0g𝑅) = (0g‘(ringLMod‘𝑅))
1816, 17eqtri 2643 . . . . . . . . . . . 12 0 = (0g‘(ringLMod‘𝑅))
1915, 18syl6eqr 2673 . . . . . . . . . . 11 ((((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) ∧ 𝑥𝐼) → (0g‘((𝐼 × {(ringLMod‘𝑅)})‘𝑥)) = 0 )
2012, 19eqtrd 2655 . . . . . . . . . 10 ((((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) ∧ 𝑥𝐼) → ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥) = 0 )
2120neeq2d 2850 . . . . . . . . 9 ((((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) ∧ 𝑥𝐼) → ((𝑘𝑥) ≠ ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥) ↔ (𝑘𝑥) ≠ 0 ))
2221rabbidva 3176 . . . . . . . 8 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → {𝑥𝐼 ∣ (𝑘𝑥) ≠ ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥)} = {𝑥𝐼 ∣ (𝑘𝑥) ≠ 0 })
23 elmapfn 7824 . . . . . . . . . 10 (𝑘 ∈ (𝑁𝑚 𝐼) → 𝑘 Fn 𝐼)
2423adantl 482 . . . . . . . . 9 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → 𝑘 Fn 𝐼)
25 fn0g 17183 . . . . . . . . . 10 0g Fn V
26 ssv 3604 . . . . . . . . . 10 ran (𝐼 × {(ringLMod‘𝑅)}) ⊆ V
27 fnco 5957 . . . . . . . . . 10 ((0g Fn V ∧ (𝐼 × {(ringLMod‘𝑅)}) Fn 𝐼 ∧ ran (𝐼 × {(ringLMod‘𝑅)}) ⊆ V) → (0g ∘ (𝐼 × {(ringLMod‘𝑅)})) Fn 𝐼)
2825, 3, 26, 27mp3an 1421 . . . . . . . . 9 (0g ∘ (𝐼 × {(ringLMod‘𝑅)})) Fn 𝐼
29 fndmdif 6277 . . . . . . . . 9 ((𝑘 Fn 𝐼 ∧ (0g ∘ (𝐼 × {(ringLMod‘𝑅)})) Fn 𝐼) → dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) = {𝑥𝐼 ∣ (𝑘𝑥) ≠ ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥)})
3024, 28, 29sylancl 693 . . . . . . . 8 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) = {𝑥𝐼 ∣ (𝑘𝑥) ≠ ((0g ∘ (𝐼 × {(ringLMod‘𝑅)}))‘𝑥)})
31 simplr 791 . . . . . . . . 9 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → 𝐼𝑊)
32 fvex 6158 . . . . . . . . . . 11 (0g𝑅) ∈ V
3316, 32eqeltri 2694 . . . . . . . . . 10 0 ∈ V
3433a1i 11 . . . . . . . . 9 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → 0 ∈ V)
35 suppvalfn 7247 . . . . . . . . 9 ((𝑘 Fn 𝐼𝐼𝑊0 ∈ V) → (𝑘 supp 0 ) = {𝑥𝐼 ∣ (𝑘𝑥) ≠ 0 })
3624, 31, 34, 35syl3anc 1323 . . . . . . . 8 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → (𝑘 supp 0 ) = {𝑥𝐼 ∣ (𝑘𝑥) ≠ 0 })
3722, 30, 363eqtr4d 2665 . . . . . . 7 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) = (𝑘 supp 0 ))
3837eleq1d 2683 . . . . . 6 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → (dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin ↔ (𝑘 supp 0 ) ∈ Fin))
39 elmapfun 7825 . . . . . . . . 9 (𝑘 ∈ (𝑁𝑚 𝐼) → Fun 𝑘)
40 id 22 . . . . . . . . 9 (𝑘 ∈ (𝑁𝑚 𝐼) → 𝑘 ∈ (𝑁𝑚 𝐼))
4133a1i 11 . . . . . . . . 9 (𝑘 ∈ (𝑁𝑚 𝐼) → 0 ∈ V)
4239, 40, 413jca 1240 . . . . . . . 8 (𝑘 ∈ (𝑁𝑚 𝐼) → (Fun 𝑘𝑘 ∈ (𝑁𝑚 𝐼) ∧ 0 ∈ V))
4342adantl 482 . . . . . . 7 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → (Fun 𝑘𝑘 ∈ (𝑁𝑚 𝐼) ∧ 0 ∈ V))
44 funisfsupp 8224 . . . . . . 7 ((Fun 𝑘𝑘 ∈ (𝑁𝑚 𝐼) ∧ 0 ∈ V) → (𝑘 finSupp 0 ↔ (𝑘 supp 0 ) ∈ Fin))
4543, 44syl 17 . . . . . 6 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → (𝑘 finSupp 0 ↔ (𝑘 supp 0 ) ∈ Fin))
4638, 45bitr4d 271 . . . . 5 (((𝑅𝑉𝐼𝑊) ∧ 𝑘 ∈ (𝑁𝑚 𝐼)) → (dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin ↔ 𝑘 finSupp 0 ))
4746rabbidva 3176 . . . 4 ((𝑅𝑉𝐼𝑊) → {𝑘 ∈ (𝑁𝑚 𝐼) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = {𝑘 ∈ (𝑁𝑚 𝐼) ∣ 𝑘 finSupp 0 })
48 eqid 2621 . . . . . . . . 9 ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼)
49 frlmbas.n . . . . . . . . . 10 𝑁 = (Base‘𝑅)
50 rlmbas 19114 . . . . . . . . . 10 (Base‘𝑅) = (Base‘(ringLMod‘𝑅))
5149, 50eqtri 2643 . . . . . . . . 9 𝑁 = (Base‘(ringLMod‘𝑅))
5248, 51pwsbas 16068 . . . . . . . 8 (((ringLMod‘𝑅) ∈ V ∧ 𝐼𝑊) → (𝑁𝑚 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)))
531, 52mpan 705 . . . . . . 7 (𝐼𝑊 → (𝑁𝑚 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)))
5453adantl 482 . . . . . 6 ((𝑅𝑉𝐼𝑊) → (𝑁𝑚 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)))
55 eqid 2621 . . . . . . . . . . 11 (Scalar‘(ringLMod‘𝑅)) = (Scalar‘(ringLMod‘𝑅))
5648, 55pwsval 16067 . . . . . . . . . 10 (((ringLMod‘𝑅) ∈ V ∧ 𝐼𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})))
571, 56mpan 705 . . . . . . . . 9 (𝐼𝑊 → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})))
5857adantl 482 . . . . . . . 8 ((𝑅𝑉𝐼𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})))
59 rlmsca 19119 . . . . . . . . . 10 (𝑅𝑉𝑅 = (Scalar‘(ringLMod‘𝑅)))
6059adantr 481 . . . . . . . . 9 ((𝑅𝑉𝐼𝑊) → 𝑅 = (Scalar‘(ringLMod‘𝑅)))
6160oveq1d 6619 . . . . . . . 8 ((𝑅𝑉𝐼𝑊) → (𝑅Xs(𝐼 × {(ringLMod‘𝑅)})) = ((Scalar‘(ringLMod‘𝑅))Xs(𝐼 × {(ringLMod‘𝑅)})))
6258, 61eqtr4d 2658 . . . . . . 7 ((𝑅𝑉𝐼𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) = (𝑅Xs(𝐼 × {(ringLMod‘𝑅)})))
6362fveq2d 6152 . . . . . 6 ((𝑅𝑉𝐼𝑊) → (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))))
6454, 63eqtrd 2655 . . . . 5 ((𝑅𝑉𝐼𝑊) → (𝑁𝑚 𝐼) = (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))))
65 rabeq 3179 . . . . 5 ((𝑁𝑚 𝐼) = (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) → {𝑘 ∈ (𝑁𝑚 𝐼) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin})
6664, 65syl 17 . . . 4 ((𝑅𝑉𝐼𝑊) → {𝑘 ∈ (𝑁𝑚 𝐼) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin} = {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin})
6747, 66eqtr3d 2657 . . 3 ((𝑅𝑉𝐼𝑊) → {𝑘 ∈ (𝑁𝑚 𝐼) ∣ 𝑘 finSupp 0 } = {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin})
689, 67syl5eq 2667 . 2 ((𝑅𝑉𝐼𝑊) → 𝐵 = {𝑘 ∈ (Base‘(𝑅Xs(𝐼 × {(ringLMod‘𝑅)}))) ∣ dom (𝑘 ∖ (0g ∘ (𝐼 × {(ringLMod‘𝑅)}))) ∈ Fin})
69 frlmval.f . . . 4 𝐹 = (𝑅 freeLMod 𝐼)
7069frlmval 20011 . . 3 ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
7170fveq2d 6152 . 2 ((𝑅𝑉𝐼𝑊) → (Base‘𝐹) = (Base‘(𝑅m (𝐼 × {(ringLMod‘𝑅)}))))
728, 68, 713eqtr4d 2665 1 ((𝑅𝑉𝐼𝑊) → 𝐵 = (Base‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  {crab 2911  Vcvv 3186  cdif 3552  wss 3555  {csn 4148   class class class wbr 4613   × cxp 5072  dom cdm 5074  ran crn 5075  ccom 5078  Fun wfun 5841   Fn wfn 5842  cfv 5847  (class class class)co 6604   supp csupp 7240  𝑚 cmap 7802  Fincfn 7899   finSupp cfsupp 8219  Basecbs 15781  Scalarcsca 15865  0gc0g 16021  Xscprds 16027  s cpws 16028  ringLModcrglmod 19088  m cdsmm 19994   freeLMod cfrlm 20009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-0g 16023  df-prds 16029  df-pws 16031  df-sra 19091  df-rgmod 19092  df-dsmm 19995  df-frlm 20010
This theorem is referenced by:  frlmelbas  20019  frlmfibas  20024  ellspd  20060  islindf4  20096  rrxbase  23084  rrxds  23089  frlmpwfi  37145
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