Theorem idlsrgplusg 31364

Description: Additive operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)

Hypotheses

Ref	Expression
idlsrgplusg.1	⊢ 𝑆 = (IDLsrg‘𝑅)
idlsrgplusg.2	⊢ ⊕ = (LSSum‘𝑅)

Assertion

Ref	Expression
idlsrgplusg	⊢ (𝑅 ∈ 𝑉 → ⊕ = (+g‘𝑆))

Proof of Theorem idlsrgplusg

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idlsrgplusg.2	. . . 4 ⊢ ⊕ = (LSSum‘𝑅)
2	1	fvexi 6731	. . 3 ⊢ ⊕ ∈ V
3		eqid 2737	. . . . 5 ⊢ ({⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (LIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ (LIdeal‘𝑅) ∧ 𝑖 ⊆ 𝑗)}⟩}) = ({⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (LIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ (LIdeal‘𝑅) ∧ 𝑖 ⊆ 𝑗)}⟩})
4	3	idlsrgstr 31361	. . . 4 ⊢ ({⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (LIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ (LIdeal‘𝑅) ∧ 𝑖 ⊆ 𝑗)}⟩}) Struct ⟨1, ;10⟩
5		plusgid 16829	. . . 4 ⊢ +g = Slot (+g‘ndx)
6		snsstp2 4730	. . . . 5 ⊢ {⟨(+g‘ndx), ⊕ ⟩} ⊆ {⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩}
7		ssun1 4086	. . . . 5 ⊢ {⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ⊆ ({⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (LIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ (LIdeal‘𝑅) ∧ 𝑖 ⊆ 𝑗)}⟩})
8	6, 7	sstri 3910	. . . 4 ⊢ {⟨(+g‘ndx), ⊕ ⟩} ⊆ ({⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (LIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ (LIdeal‘𝑅) ∧ 𝑖 ⊆ 𝑗)}⟩})
9	4, 5, 8	strfv 16754	. . 3 ⊢ ( ⊕ ∈ V → ⊕ = (+g‘({⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (LIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ (LIdeal‘𝑅) ∧ 𝑖 ⊆ 𝑗)}⟩})))
10	2, 9	ax-mp 5	. 2 ⊢ ⊕ = (+g‘({⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (LIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ (LIdeal‘𝑅) ∧ 𝑖 ⊆ 𝑗)}⟩}))
11		idlsrgplusg.1	. . . 4 ⊢ 𝑆 = (IDLsrg‘𝑅)
12		eqid 2737	. . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
13		eqid 2737	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
14		eqid 2737	. . . . 5 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
15	12, 1, 13, 14	idlsrgval 31362	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (LIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ (LIdeal‘𝑅) ∧ 𝑖 ⊆ 𝑗)}⟩}))
16	11, 15	syl5eq 2790	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = ({⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (LIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ (LIdeal‘𝑅) ∧ 𝑖 ⊆ 𝑗)}⟩}))
17	16	fveq2d 6721	. 2 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑆) = (+g‘({⟨(Base‘ndx), (LIdeal‘𝑅)⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ (LIdeal‘𝑅), 𝑗 ∈ (LIdeal‘𝑅) ↦ ((RSpan‘𝑅)‘(𝑖(LSSum‘(mulGrp‘𝑅))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (LIdeal‘𝑅) ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ (LIdeal‘𝑅) ∧ 𝑖 ⊆ 𝑗)}⟩})))
18	10, 17	eqtr4id 2797	1 ⊢ (𝑅 ∈ 𝑉 → ⊕ = (+g‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  {crab 3065  Vcvv 3408   ∪ cun 3864   ⊆ wss 3866  {csn 4541  {cpr 4543  {ctp 4545  ⟨cop 4547  {copab 5115   ↦ cmpt 5135  ran crn 5552  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  0cc0 10729  1c1 10730  ;cdc 12293  ndxcnx 16744  Basecbs 16760  +_gcplusg 16802  ._rcmulr 16803  TopSetcts 16808  lecple 16809  LSSumclsm 19023  mulGrpcmgp 19504  LIdealclidl 20207  RSpancrsp 20208  IDLsrgcidlsrg 31359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-fz 13096  df-struct 16700  df-slot 16735  df-ndx 16745  df-base 16761  df-plusg 16815  df-mulr 16816  df-tset 16821  df-ple 16822  df-idlsrg 31360

This theorem is referenced by:  idlsrg0g  31365  idlsrgmnd  31373  idlsrgcmnd  31374