Theorem idlsrgplusg 33660

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 6875	. . 3 ⊢ ⊕ ∈ V
3		eqid 2761	. . . . 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 33657	. . . 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 17294	. . . 4 ⊢ +g = Slot (+g‘ndx)
6		snsstp2 4774	. . . . 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 4130	. . . . 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 3945	. . . 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 2761	. . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
13		eqid 2761	. . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
14		eqid 2761	. . . . 5 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
15	12, 1, 13, 14	idlsrgval 33658	. . . 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 2808	. . 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 6865	. 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 2815	1 ⊢ (𝑅 ∈ 𝑉 → ⊕ = (+g‘𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453   ∪ cun 3902   ⊆ wss 3904  {csn 4581  {cpr 4583  {ctp 4585  ⟨cop 4587  {copab 5161   ↦ cmpt 5180  ran crn 5646  ‘cfv 6515  (class class class)co 7390   ∈ cmpo 7392  0cc0 11068  1c1 11069  ;cdc 12683  ndxcnx 17210  Basecbs 17226  +_gcplusg 17267  ._rcmulr 17268  TopSetcts 17273  lecple 17274  LSSumclsm 19655  mulGrpcmgp 20167  LIdealclidl 21254  RSpancrsp 21255  IDLsrgcidlsrg 33655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-er 8671  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11213  df-mnf 11214  df-xr 11215  df-ltxr 11216  df-le 11217  df-sub 11411  df-neg 11412  df-nn 12206  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12477  df-z 12564  df-dec 12684  df-uz 12835  df-fz 13508  df-struct 17164  df-slot 17199  df-ndx 17211  df-base 17227  df-plusg 17280  df-mulr 17281  df-tset 17286  df-ple 17287  df-idlsrg 33656

This theorem is referenced by:  idlsrg0g  33661  idlsrgmnd  33669  idlsrgcmnd  33670