Theorem idlsrgval 33921

Description: Lemma for idlsrgbas 33922 through idlsrgtset 33926. (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 3474	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		fvexd 6897	. . . 4 ⊢ (𝑟 = 𝑅 → (LIdeal‘𝑟) ∈ V)
3		simpr 490	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → 𝑏 = (LIdeal‘𝑟))
4		simpl 488	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → 𝑟 = 𝑅)
5	4	fveq2d 6886	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LIdeal‘𝑟) = (LIdeal‘𝑅))
6	3, 5	eqtrd 2797	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → 𝑏 = (LIdeal‘𝑅))
7		idlsrgval.1	. . . . . . . 8 ⊢ 𝐼 = (LIdeal‘𝑅)
8	6, 7	eqtr4di 2815	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → 𝑏 = 𝐼)
9	8	opeq2d 4843	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐼⟩)
10	4	fveq2d 6886	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘𝑟) = (LSSum‘𝑅))
11		idlsrgval.2	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑅)
12	10, 11	eqtr4di 2815	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘𝑟) = ⊕ )
13	12	opeq2d 4843	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ⟨(+g‘ndx), (LSSum‘𝑟)⟩ = ⟨(+g‘ndx), ⊕ ⟩)
14	4	fveq2d 6886	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (RSpan‘𝑟) = (RSpan‘𝑅))
15	4	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
16		idlsrgval.3	. . . . . . . . . . . . 13 ⊢ 𝐺 = (mulGrp‘𝑅)
17	15, 16	eqtr4di 2815	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (mulGrp‘𝑟) = 𝐺)
18	17	fveq2d 6886	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘(mulGrp‘𝑟)) = (LSSum‘𝐺))
19		idlsrgval.4	. . . . . . . . . . 11 ⊢ ⊗ = (LSSum‘𝐺)
20	18, 19	eqtr4di 2815	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘(mulGrp‘𝑟)) = ⊗ )
21	20	oveqd 7434	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (𝑖(LSSum‘(mulGrp‘𝑟))𝑗) = (𝑖 ⊗ 𝑗))
22	14, 21	fveq12d 6889	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)) = ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))
23	8, 8, 22	mpoeq123dv 7492	. . . . . . 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 4843	. . . . . 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 4712	. . . . 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 3429	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗} = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
27	8, 26	mpteq12dv 5196	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}))
28	27	rneqd 5926	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}))
29	28	opeq2d 4843	. . . . . 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 3966	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ({𝑖, 𝑗} ⊆ 𝑏 ↔ {𝑖, 𝑗} ⊆ 𝐼))
31	30	anbi1d 643	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗) ↔ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)))
32	31	opabbidv 5175	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)})
33	32	opeq2d 4843	. . . . . 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 4705	. . . . 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 4119	. . . 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 3886	. . 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 33919	. . 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 7751	. . . 4 ⊢ {⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∈ V
39		prex 5407	. . . 4 ⊢ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩} ∈ V
40	38, 39	unex 7750	. . 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 6990	. 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 18	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)}⟩}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453  ⦋csb 3850   ∪ cun 3900   ⊆ wss 3902  {cpr 4589  {ctp 4591  ⟨cop 4593  {copab 5171   ↦ cmpt 5190  ran crn 5660  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  ndxcnx 17291  Basecbs 17307  +_gcplusg 17348  ._rcmulr 17349  TopSetcts 17354  lecple 17355  LSSumclsm 19767  mulGrpcmgp 20279  LIdealclidl 21399  RSpancrsp 21400  IDLsrgcidlsrg 33918

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-idlsrg 33919

This theorem is used by:  idlsrgbas  33922  idlsrgplusg  33923  idlsrgmulr  33925  idlsrgtset  33926