Theorem idlsrgval 33474

Description: Lemma for idlsrgbas 33475 through idlsrgtset 33479. (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 3468	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		fvexd 6873	. . . 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 6862	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LIdeal‘𝑟) = (LIdeal‘𝑅))
6	3, 5	eqtrd 2764	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → 𝑏 = (LIdeal‘𝑅))
7		idlsrgval.1	. . . . . . . 8 ⊢ 𝐼 = (LIdeal‘𝑅)
8	6, 7	eqtr4di 2782	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → 𝑏 = 𝐼)
9	8	opeq2d 4844	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐼⟩)
10	4	fveq2d 6862	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘𝑟) = (LSSum‘𝑅))
11		idlsrgval.2	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑅)
12	10, 11	eqtr4di 2782	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘𝑟) = ⊕ )
13	12	opeq2d 4844	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ⟨(+g‘ndx), (LSSum‘𝑟)⟩ = ⟨(+g‘ndx), ⊕ ⟩)
14	4	fveq2d 6862	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (RSpan‘𝑟) = (RSpan‘𝑅))
15	4	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (mulGrp‘𝑟) = (mulGrp‘𝑅))
16		idlsrgval.3	. . . . . . . . . . . . 13 ⊢ 𝐺 = (mulGrp‘𝑅)
17	15, 16	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (mulGrp‘𝑟) = 𝐺)
18	17	fveq2d 6862	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘(mulGrp‘𝑟)) = (LSSum‘𝐺))
19		idlsrgval.4	. . . . . . . . . . 11 ⊢ ⊗ = (LSSum‘𝐺)
20	18, 19	eqtr4di 2782	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (LSSum‘(mulGrp‘𝑟)) = ⊗ )
21	20	oveqd 7404	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (𝑖(LSSum‘(mulGrp‘𝑟))𝑗) = (𝑖 ⊗ 𝑗))
22	14, 21	fveq12d 6865	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)) = ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))
23	8, 8, 22	mpoeq123dv 7464	. . . . . . 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 4844	. . . . . 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 3421	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗} = {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})
27	8, 26	mpteq12dv 5194	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗}) = (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}))
28	27	rneqd 5902	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗}) = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗}))
29	28	opeq2d 4844	. . . . . 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 3979	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → ({𝑖, 𝑗} ⊆ 𝑏 ↔ {𝑖, 𝑗} ⊆ 𝐼))
31	30	anbi1d 631	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → (({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗) ↔ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)))
32	31	opabbidv 5173	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (LIdeal‘𝑟)) → {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)})
33	32	opeq2d 4844	. . . . . 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 4132	. . . 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 3898	. . 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 33472	. . 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 7722	. . . 4 ⊢ {⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∈ V
39		prex 5392	. . . 4 ⊢ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩} ∈ V
40	38, 39	unex 7720	. . 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 6968	. 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 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447  ⦋csb 3862   ∪ cun 3912   ⊆ wss 3914  {cpr 4591  {ctp 4593  ⟨cop 4595  {copab 5169   ↦ cmpt 5188  ran crn 5639  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  ndxcnx 17163  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  TopSetcts 17226  lecple 17227  LSSumclsm 19564  mulGrpcmgp 20049  LIdealclidl 21116  RSpancrsp 21117  IDLsrgcidlsrg 33471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-idlsrg 33472

This theorem is referenced by:  idlsrgbas  33475  idlsrgplusg  33476  idlsrgmulr  33478  idlsrgtset  33479