Theorem idlsrgplusg 33466

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 6889	. . 3 ⊢ ⊕ ∈ V
3		eqid 2735	. . . . 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 33463	. . . 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 17296	. . . 4 ⊢ +g = Slot (+g‘ndx)
6		snsstp2 4793	. . . . 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 4153	. . . . 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 3968	. . . 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 17220	. . 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 2735	. . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
13		eqid 2735	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
14		eqid 2735	. . . . 5 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
15	12, 1, 13, 14	idlsrgval 33464	. . . 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	eqtrid 2782	. . 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 6879	. 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 2789	1 ⊢ (𝑅 ∈ 𝑉 → ⊕ = (+g‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3415  Vcvv 3459   ∪ cun 3924   ⊆ wss 3926  {csn 4601  {cpr 4603  {ctp 4605  ⟨cop 4607  {copab 5181   ↦ cmpt 5201  ran crn 5655  ‘cfv 6530  (class class class)co 7403   ∈ cmpo 7405  0cc0 11127  1c1 11128  ;cdc 12706  ndxcnx 17210  Basecbs 17226  +_gcplusg 17269  ._rcmulr 17270  TopSetcts 17275  lecple 17276  LSSumclsm 19613  mulGrpcmgp 20098  LIdealclidl 21165  RSpancrsp 21166  IDLsrgcidlsrg 33461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12851  df-fz 13523  df-struct 17164  df-slot 17199  df-ndx 17211  df-base 17227  df-plusg 17282  df-mulr 17283  df-tset 17288  df-ple 17289  df-idlsrg 33462

This theorem is referenced by:  idlsrg0g  33467  idlsrgmnd  33475  idlsrgcmnd  33476