Theorem opsrle 22014

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 2737	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑇 ⊆ (𝐼 × 𝐼))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	opsrval 22013	. . . 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 6846	. . 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 7402	. . . 4 ⊢ 𝑆 ∈ V
16	3	fvexi 6856	. . . . . 6 ⊢ 𝐵 ∈ V
17	16, 16	xpex 7708	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
18		vex 3446	. . . . . . . . 9 ⊢ 𝑥 ∈ V
19		vex 3446	. . . . . . . . 9 ⊢ 𝑦 ∈ V
20	18, 19	prss 4778	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21	20	anbi1i 625	. . . . . . 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 5167	. . . . . 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 5724	. . . . . 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 3983	. . . . 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 5269	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ (∃𝑧 ∈ 𝐷 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐷 (𝑤𝐶𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))} ∈ V
26		pleid 17299	. . . . 5 ⊢ le = Slot (le‘ndx)
27	26	setsid 17146	. . . 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 693	. . 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 2797	. 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 22012	. . . . . . . . . 10 ⊢ Rel dom ordPwSer
31	30	ovprc 7406	. . . . . . . . 9 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 ordPwSer 𝑅) = ∅)
32	31	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐼 ordPwSer 𝑅) = ∅)
33	32	fveq1d 6844	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → ((𝐼 ordPwSer 𝑅)‘𝑇) = (∅‘𝑇))
34	2, 33	eqtrid 2784	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = (∅‘𝑇))
35		0fv 6883	. . . . . 6 ⊢ (∅‘𝑇) = ∅
36	34, 35	eqtrdi 2788	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑂 = ∅)
37	36	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (le‘𝑂) = (le‘∅))
38	26	str0 17128	. . . 4 ⊢ ∅ = (le‘∅)
39	37, 14, 38	3eqtr4g 2797	. . 3 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → ≤ = ∅)
40		reldmpsr 21882	. . . . . . . . . . 11 ⊢ Rel dom mPwSer
41	40	ovprc 7406	. . . . . . . . . 10 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
42	41	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐼 mPwSer 𝑅) = ∅)
43	1, 42	eqtrid 2784	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝑆 = ∅)
44	43	fveq2d 6846	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (Base‘𝑆) = (Base‘∅))
45		base0 17153	. . . . . . 7 ⊢ ∅ = (Base‘∅)
46	44, 3, 45	3eqtr4g 2797	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → 𝐵 = ∅)
47	46	xpeq2d 5662	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐵 × 𝐵) = (𝐵 × ∅))
48		xp0 5732	. . . . 5 ⊢ (𝐵 × ∅) = ∅
49	47, 48	eqtrdi 2788	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝐼 ∈ V ∧ 𝑅 ∈ V)) → (𝐵 × 𝐵) = ∅)
50		sseq0 4357	. . . 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 588	. . 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 2775	. 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 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  {cpr 4584  ⟨cop 4588   class class class wbr 5100  {copab 5162   × cxp 5630  ^◡ccnv 5631   “ cima 5635  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775  Fincfn 8895  ℕcn 12157  ℕ₀cn0 12413   sSet csts 17102  ndxcnx 17132  Basecbs 17148  lecple 17196  ltcplt 18243   mPwSer cmps 21872   <_bag cltb 21875   ordPwSer copws 21876

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-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  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-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-ltxr 11183  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-dec 12620  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ple 17209  df-psr 21877  df-opsr 21881

This theorem is referenced by:  opsrval2  22015  opsrtoslem1  22022