Theorem idlsrgval 33595

Description: Lemma for idlsrgbas 33596 through idlsrgtset 33600. (Contributed by Thierry Arnoux, 1-Jun-2024.)

Hypotheses

Ref	Expression
idlsrgval.1	⊢ 𝐼 = (LIdeal‘𝑅)
idlsrgval.2	⊢ ⊕ = (LSSum‘𝑅)
idlsrgval.3	⊢ 𝐺 = (mulGrp‘𝑅)
idlsrgval.4	⊢ ⊗ = (LSSum‘𝐺)

Assertion

Ref	Expression
idlsrgval	⊢ (𝑅 ∈ 𝑉 → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩}))

Distinct variable groups:   𝑖,𝐼,𝑗   𝑅,𝑖,𝑗

Allowed substitution hints:   ⊕ (𝑖,𝑗)   ⊗ (𝑖,𝑗)   𝐺(𝑖,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem idlsrgval

Dummy variables 𝑏 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3463	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		fvexd 6857	. . . 4 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) ∈ V)
3		simpr 484	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → 𝑏 = (LIdeal‘𝑟))
4		simpl 482	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → 𝑟 = 𝑅)
5	4	fveq2d 6846	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LIdeal‘𝑟) = (LIdeal‘𝑅))
6	3, 5	eqtrd 2772	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → 𝑏 = (LIdeal‘𝑅))
7		idlsrgval.1	. . . . . . . 8 ⊢ 𝐼 = (LIdeal‘𝑅)
8	6, 7	eqtr4di 2790	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → 𝑏 = 𝐼)
9	8	opeq2d 4838	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐼⟩)
10	4	fveq2d 6846	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘𝑟) = (LSSum‘𝑅))
11		idlsrgval.2	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑅)
12	10, 11	eqtr4di 2790	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘𝑟) = ⊕ )
13	12	opeq2d 4838	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ⟨(+g‘ndx), (LSSum‘𝑟)⟩ = ⟨(+g‘ndx), ⊕ ⟩)
14	4	fveq2d 6846	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (RSpan‘𝑟) = (RSpan‘𝑅))
15	4	fveq2d 6846	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
16		idlsrgval.3	. . . . . . . . . . . . 13 ⊢ 𝐺 = (mulGrp‘𝑅)
17	15, 16	eqtr4di 2790	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (mulGrp‘𝑟) = 𝐺)
18	17	fveq2d 6846	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘(mulGrp‘𝑟)) = (LSSum‘𝐺))
19		idlsrgval.4	. . . . . . . . . . 11 ⊢ ⊗ = (LSSum‘𝐺)
20	18, 19	eqtr4di 2790	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘(mulGrp‘𝑟)) = ⊗ )
21	20	oveqd 7385	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (𝑖(LSSum‘(mulGrp‘𝑟))𝑗) = (𝑖 ⊗ 𝑗))
22	14, 21	fveq12d 6849	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)) = ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))
23	8, 8, 22	mpoeq123dv 7443	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗))) = (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗))))
24	23	opeq2d 4838	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ⟨(.r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩ = ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩)
25	9, 13, 24	tpeq123d 4707	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} = {⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩})
26	8	rabeqdv 3416	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗} = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
27	8, 26	mpteq12dv 5187	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}))
28	27	rneqd 5895	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}))
29	28	opeq2d 4838	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ⟨(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗})⟩ = ⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩)
30	8	sseq2d 3968	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ({𝑖, 𝑗} ⊆ 𝑏 ↔ {𝑖, 𝑗} ⊆ 𝐼))
31	30	anbi1d 632	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗) ↔ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)))
32	31	opabbidv 5166	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)})
33	32	opeq2d 4838	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}⟩ = ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩)
34	29, 33	preq12d 4700	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}⟩} = {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩})
35	25, 34	uneq12d 4123	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}⟩}) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩}))
36	2, 35	csbied 3887	. . 3 ⊢ (𝑟 = 𝑅 → ⦋(LIdeal‘𝑟) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}⟩}) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩}))
37		df-idlsrg 33593	. . 3 ⊢ IDLsrg = (𝑟 ∈ V ↦ ⦋(LIdeal‘𝑟) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}⟩}))
38		tpex 7701	. . . 4 ⊢ {⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∈ V
39		prex 5384	. . . 4 ⊢ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩} ∈ V
40	38, 39	unex 7699	. . 3 ⊢ ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩}) ∈ V
41	36, 37, 40	fvmpt 6949	. 2 ⊢ (𝑅 ∈ V → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩}))
42	1, 41	syl 17	1 ⊢ (𝑅 ∈ 𝑉 → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442  ⦋csb 3851   ∪ cun 3901   ⊆ wss 3903  {cpr 4584  {ctp 4586  ⟨cop 4588  {copab 5162   ↦ cmpt 5181  ran crn 5633  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  ndxcnx 17132  Basecbs 17148  +_gcplusg 17189  ._rcmulr 17190  TopSetcts 17195  lecple 17196  LSSumclsm 19575  mulGrpcmgp 20087  LIdealclidl 21173  RSpancrsp 21174  IDLsrgcidlsrg 33592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-idlsrg 33593

This theorem is referenced by:  idlsrgbas  33596  idlsrgplusg  33597  idlsrgmulr  33599  idlsrgtset  33600