Theorem isinftm 31154

Description: Express 𝑥 is infinitesimal with respect to 𝑦 for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Hypotheses

Ref	Expression
inftm.b	⊢ 𝐵 = (Base‘𝑊)
inftm.0	⊢ 0 = (0g‘𝑊)
inftm.x	⊢ · = (.g‘𝑊)
inftm.l	⊢ < = (lt‘𝑊)

Assertion

Ref	Expression
isinftm	⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))

Distinct variable groups:   𝑛,𝑊   𝑛,𝑋   𝑛,𝑌

Allowed substitution hints:   𝐵(𝑛)   < (𝑛)   · (𝑛)   𝑉(𝑛)   0 (𝑛)

Proof of Theorem isinftm

Dummy variables 𝑥 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2825	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
2		eleq1 2825	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ 𝐵 ↔ 𝑌 ∈ 𝐵))
3	1, 2	bi2anan9 639	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)))
4		simpl 486	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
5	4	breq2d 5065	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ( 0 < 𝑥 ↔ 0 < 𝑋))
6	4	oveq2d 7229	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑛 · 𝑥) = (𝑛 · 𝑋))
7		simpr 488	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
8	6, 7	breq12d 5066	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑛 · 𝑥) < 𝑦 ↔ (𝑛 · 𝑋) < 𝑌))
9	8	ralbidv 3118	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))
10	5, 9	anbi12d 634	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦) ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
11	3, 10	anbi12d 634	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦)) ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
12		eqid 2737	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}
13	11, 12	brabga 5415	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
14	13	3adant1 1132	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
15		elex 3426	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
16	15	3ad2ant1 1135	. . . 4 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑊 ∈ V)
17		fveq2 6717	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
18		inftm.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑊)
19	17, 18	eqtr4di 2796	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
20	19	eleq2d 2823	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ 𝐵))
21	19	eleq2d 2823	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦 ∈ 𝐵))
22	20, 21	anbi12d 634	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
23		fveq2 6717	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
24		inftm.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑊)
25	23, 24	eqtr4di 2796	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
26		fveq2 6717	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
27		inftm.l	. . . . . . . . . 10 ⊢ < = (lt‘𝑊)
28	26, 27	eqtr4di 2796	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (lt‘𝑤) = < )
29		eqidd 2738	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
30	25, 28, 29	breq123d 5067	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((0g‘𝑤)(lt‘𝑤)𝑥 ↔ 0 < 𝑥))
31		fveq2 6717	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (.g‘𝑤) = (.g‘𝑊))
32		inftm.x	. . . . . . . . . . . 12 ⊢ · = (.g‘𝑊)
33	31, 32	eqtr4di 2796	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (.g‘𝑤) = · )
34	33	oveqd 7230	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑛(.g‘𝑤)𝑥) = (𝑛 · 𝑥))
35		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑦 = 𝑦)
36	34, 28, 35	breq123d 5067	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛 · 𝑥) < 𝑦))
37	36	ralbidv 3118	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))
38	30, 37	anbi12d 634	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦) ↔ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦)))
39	22, 38	anbi12d 634	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))))
40	39	opabbidv 5119	. . . . 5 ⊢ (𝑤 = 𝑊 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))})
41		df-inftm 31151	. . . . 5 ⊢ ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦))})
42	18	fvexi 6731	. . . . . . 7 ⊢ 𝐵 ∈ V
43	42, 42	xpex 7538	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
44		opabssxp 5640	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ⊆ (𝐵 × 𝐵)
45	43, 44	ssexi 5215	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ∈ V
46	40, 41, 45	fvmpt 6818	. . . 4 ⊢ (𝑊 ∈ V → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))})
47	16, 46	syl 17	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))})
48	47	breqd 5064	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ 𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌))
49		3simpc 1152	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
50	49	biantrurd 536	. 2 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌) ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
51	14, 48, 50	3bitr4d 314	1 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408   class class class wbr 5053  {copab 5115   × cxp 5549  ‘cfv 6380  (class class class)co 7213  ℕcn 11830  Basecbs 16760  0_gc0g 16944  ltcplt 17815  ._gcmg 18488  ⋘cinftm 31149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-inftm 31151

This theorem is referenced by:  pnfinf  31156  isarchi2  31158