Theorem idlsrgplusg 33736

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 6893	. . 3 ⊢ ⊕ ∈ V
3		eqid 2769	. . . . 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 33733	. . . 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 17333	. . . 4 ⊢ +g = Slot (+g‘ndx)
6		snsstp2 4784	. . . . 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 4139	. . . . 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 3954	. . . 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 17259	. . 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 2769	. . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
13		eqid 2769	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
14		eqid 2769	. . . . 5 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
15	12, 1, 13, 14	idlsrgval 33734	. . . 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 2816	. . 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 6883	. 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 2823	1 ⊢ (𝑅 ∈ 𝑉 → ⊕ = (+g‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  {csn 4591  {cpr 4593  {ctp 4595  ⟨cop 4597  {copab 5174   ↦ cmpt 5193  ran crn 5660  ‘cfv 6533  (class class class)co 7408   ∈ cmpo 7410  0cc0 11096  1c1 11097  ;cdc 12707  ndxcnx 17249  Basecbs 17265  +_gcplusg 17306  ._rcmulr 17307  TopSetcts 17312  lecple 17313  LSSumclsm 19700  mulGrpcmgp 20212  LIdealclidl 21304  RSpancrsp 21305  IDLsrgcidlsrg 33731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-z 12588  df-dec 12708  df-uz 12859  df-fz 13532  df-struct 17203  df-slot 17238  df-ndx 17250  df-base 17266  df-plusg 17319  df-mulr 17320  df-tset 17325  df-ple 17326  df-idlsrg 33732

This theorem is referenced by:  idlsrg0g  33737  idlsrgmnd  33745  idlsrgcmnd  33746