Theorem opsrle 20851

Description: An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
opsrle.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
opsrle.o	⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
opsrle.b	⊢ 𝐵 = (Base‘𝑆)
opsrle.q	⊢ < = (lt‘𝑅)
opsrle.c	⊢ 𝐶 = (𝑇 <bag 𝐼)
opsrle.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
opsrle.l	⊢ ≤ = (le‘𝑂)
opsrle.t	⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))

Assertion

Ref	Expression
opsrle	⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))})

Distinct variable groups:   𝑥,𝑦,𝐵   𝑧,𝑤,𝐷   𝑤,ℎ,𝑥,𝑦,𝑧,𝐼   𝑤,𝑅,𝑥,𝑦,𝑧   𝜑,𝑤,𝑥,𝑦,𝑧   𝑤,𝑇,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(ℎ)   𝐵(𝑧,𝑤,ℎ)   𝐶(𝑥,𝑦,𝑧,𝑤,ℎ)   𝐷(𝑥,𝑦,ℎ)   𝑅(ℎ)   𝑆(𝑥,𝑦,𝑧,𝑤,ℎ)   < (𝑥,𝑦,𝑧,𝑤,ℎ)   𝑇(ℎ)   ≤ (𝑥,𝑦,𝑧,𝑤,ℎ)   𝑂(𝑥,𝑦,𝑧,𝑤,ℎ)

Proof of Theorem opsrle

Step	Hyp	Ref	Expression
1		opsrle.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		opsrle.o	. . . . 5 ⊢ 𝑂 = ((𝐼 ordPwSer 𝑅)‘𝑇)
3		opsrle.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑆)
4		opsrle.q	. . . . 5 ⊢ < = (lt‘𝑅)
5		opsrle.c	. . . . 5 ⊢ 𝐶 = (𝑇 <bag 𝐼)
6		opsrle.d	. . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
7		eqid 2738	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}
8		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐼 ∈ V)
9		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑅 ∈ V)
10		opsrle.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝐼 × 𝐼))
11	10	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑇 ⊆ (𝐼 × 𝐼))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	opsrval 20850	. . . 4 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = (𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩))
13	12	fveq2d 6672	. . 3 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (le‘𝑂) = (le‘(𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
14		opsrle.l	. . 3 ⊢ ≤ = (le‘𝑂)
15	1	ovexi 7198	. . . 4 ⊢ 𝑆 ∈ V
16	3	fvexi 6682	. . . . . 6 ⊢ 𝐵 ∈ V
17	16, 16	xpex 7488	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
18		vex 3401	. . . . . . . . 9 ⊢ 𝑥 ∈ V
19		vex 3401	. . . . . . . . 9 ⊢ 𝑦 ∈ V
20	18, 19	prss 4705	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21	20	anbi1i 627	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦)) ↔ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦)))
22	21	opabbii 5094	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}
23		opabssxp 5608	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} ⊆ (𝐵 × 𝐵)
24	22, 23	eqsstrri 3910	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} ⊆ (𝐵 × 𝐵)
25	17, 24	ssexi 5187	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} ∈ V
26		pleid 16763	. . . . 5 ⊢ le = Slot (le‘ndx)
27	26	setsid 16634	. . . 4 ⊢ ((𝑆 ∈ V ∧ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = (le‘(𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩)))
28	15, 25, 27	mp2an 692	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = (le‘(𝑆 sSet ⟨(le‘ndx), {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}⟩))
29	13, 14, 28	3eqtr4g 2798	. 2 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))})
30		reldmopsr 20849	. . . . . . . . . 10 ⊢ Rel dom ordPwSer
31	30	ovprc 7202	. . . . . . . . 9 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 ordPwSer 𝑅) = ∅)
32	31	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐼 ordPwSer 𝑅) = ∅)
33	32	fveq1d 6670	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → ((𝐼 ordPwSer 𝑅)‘𝑇) = (∅‘𝑇))
34	2, 33	syl5eq 2785	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = (∅‘𝑇))
35		0fv 6707	. . . . . 6 ⊢ (∅‘𝑇) = ∅
36	34, 35	eqtrdi 2789	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = ∅)
37	36	fveq2d 6672	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (le‘𝑂) = (le‘∅))
38	26	str0 16631	. . . 4 ⊢ ∅ = (le‘∅)
39	37, 14, 38	3eqtr4g 2798	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → ≤ = ∅)
40		reldmpsr 20720	. . . . . . . . . . 11 ⊢ Rel dom mPwSer
41	40	ovprc 7202	. . . . . . . . . 10 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
42	41	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐼 mPwSer 𝑅) = ∅)
43	1, 42	syl5eq 2785	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑆 = ∅)
44	43	fveq2d 6672	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (Base‘𝑆) = (Base‘∅))
45		base0 16632	. . . . . . 7 ⊢ ∅ = (Base‘∅)
46	44, 3, 45	3eqtr4g 2798	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐵 = ∅)
47	46	xpeq2d 5549	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐵 × 𝐵) = (𝐵 × ∅))
48		xp0 5984	. . . . 5 ⊢ (𝐵 × ∅) = ∅
49	47, 48	eqtrdi 2789	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐵 × 𝐵) = ∅)
50		sseq0 4285	. . . 4 ⊢ (({⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} ⊆ (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) = ∅) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = ∅)
51	24, 49, 50	sylancr 590	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} = ∅)
52	39, 51	eqtr4d 2776	. 2 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))})
53	29, 52	pm2.61dan 813	1 ⊢ (𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 846   = wceq 1542   ∈ wcel 2113  ∀wral 3053  ∃wrex 3054  {crab 3057  Vcvv 3397   ⊆ wss 3841  ∅c0 4209  {cpr 4515  ⟨cop 4519   class class class wbr 5027  {copab 5089   × cxp 5517  ^◡ccnv 5518   “ cima 5522  ‘cfv 6333  (class class class)co 7164   ↑_m cmap 8430  Fincfn 8548  ℕcn 11709  ℕ₀cn0 11969  ndxcnx 16576   sSet csts 16577  Basecbs 16579  lecple 16668  ltcplt 17660   mPwSer cmps 20710   <_bag cltb 20713   ordPwSer copws 20714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-er 8313  df-en 8549  df-dom 8550  df-sdom 8551  df-pnf 10748  df-mnf 10749  df-ltxr 10751  df-nn 11710  df-2 11772  df-3 11773  df-4 11774  df-5 11775  df-6 11776  df-7 11777  df-8 11778  df-9 11779  df-dec 12173  df-ndx 16582  df-slot 16583  df-base 16585  df-sets 16586  df-ple 16681  df-psr 20715  df-opsr 20719

This theorem is referenced by:  opsrval2  20852  opsrtoslem1  20859