Theorem idlsrgplusg 31552

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 6770	. . 3 ⊢ ⊕ ∈ V
3		eqid 2738	. . . . 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 31549	. . . 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 16915	. . . 4 ⊢ +g = Slot (+g‘ndx)
6		snsstp2 4747	. . . . 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 4102	. . . . 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 3926	. . . 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 16833	. . 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 2738	. . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
13		eqid 2738	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
14		eqid 2738	. . . . 5 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
15	12, 1, 13, 14	idlsrgval 31550	. . . 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 2791	. . 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 6760	. 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 2798	1 ⊢ (𝑅 ∈ 𝑉 → ⊕ = (+g‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  {csn 4558  {cpr 4560  {ctp 4562  ⟨cop 4564  {copab 5132   ↦ cmpt 5153  ran crn 5581  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  0cc0 10802  1c1 10803  ;cdc 12366  ndxcnx 16822  Basecbs 16840  +_gcplusg 16888  ._rcmulr 16889  TopSetcts 16894  lecple 16895  LSSumclsm 19154  mulGrpcmgp 19635  LIdealclidl 20347  RSpancrsp 20348  IDLsrgcidlsrg 31547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-struct 16776  df-slot 16811  df-ndx 16823  df-base 16841  df-plusg 16901  df-mulr 16902  df-tset 16907  df-ple 16908  df-idlsrg 31548

This theorem is referenced by:  idlsrg0g  31553  idlsrgmnd  31561  idlsrgcmnd  31562