Theorem isinftm 33097

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 2821	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵))
2		eleq1 2821	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ 𝐵 ↔ 𝑌 ∈ 𝐵))
3	1, 2	bi2anan9 638	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)))
4		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
5	4	breq2d 5128	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ( 0 < 𝑥 ↔ 0 < 𝑋))
6	4	oveq2d 7415	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑛 · 𝑥) = (𝑛 · 𝑋))
7		simpr 484	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
8	6, 7	breq12d 5129	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑛 · 𝑥) < 𝑦 ↔ (𝑛 · 𝑋) < 𝑌))
9	8	ralbidv 3161	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))
10	5, 9	anbi12d 632	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦) ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
11	3, 10	anbi12d 632	. . . 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 2734	. . . 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 5506	. . 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 1130	. 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 3478	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V)
16	15	3ad2ant1 1133	. . . 4 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑊 ∈ V)
17		fveq2 6872	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
18		inftm.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑊)
19	17, 18	eqtr4di 2787	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
20	19	eleq2d 2819	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥 ∈ 𝐵))
21	19	eleq2d 2819	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦 ∈ 𝐵))
22	20, 21	anbi12d 632	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
23		fveq2 6872	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
24		inftm.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑊)
25	23, 24	eqtr4di 2787	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
26		fveq2 6872	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
27		inftm.l	. . . . . . . . . 10 ⊢ < = (lt‘𝑊)
28	26, 27	eqtr4di 2787	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (lt‘𝑤) = < )
29		eqidd 2735	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
30	25, 28, 29	breq123d 5130	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((0g‘𝑤)(lt‘𝑤)𝑥 ↔ 0 < 𝑥))
31		fveq2 6872	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (.g‘𝑤) = (.g‘𝑊))
32		inftm.x	. . . . . . . . . . . 12 ⊢ · = (.g‘𝑊)
33	31, 32	eqtr4di 2787	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (.g‘𝑤) = · )
34	33	oveqd 7416	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑛(.g‘𝑤)𝑥) = (𝑛 · 𝑥))
35		eqidd 2735	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑦 = 𝑦)
36	34, 28, 35	breq123d 5130	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛 · 𝑥) < 𝑦))
37	36	ralbidv 3161	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))
38	30, 37	anbi12d 632	. . . . . . 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 632	. . . . . 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 5182	. . . . 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 33094	. . . . 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 6886	. . . . . . 7 ⊢ 𝐵 ∈ V
43	42, 42	xpex 7741	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
44		opabssxp 5744	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ⊆ (𝐵 × 𝐵)
45	43, 44	ssexi 5289	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ∈ V
46	40, 41, 45	fvmpt 6982	. . . 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 5127	. 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 1150	. . 3 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
50	49	biantrurd 532	. 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 311	1 ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  Vcvv 3457   class class class wbr 5116  {copab 5178   × cxp 5649  ‘cfv 6527  (class class class)co 7399  ℕcn 12232  Basecbs 17213  0_gc0g 17438  ltcplt 18305  ._gcmg 19035  ⋘cinftm 33092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6480  df-fun 6529  df-fv 6535  df-ov 7402  df-inftm 33094

This theorem is referenced by:  pnfinf  33099  isarchi2  33101