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Theorem isinftm 29066
 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 = (0g𝑊)
inftm.x · = (.g𝑊)
inftm.l < = (lt‘𝑊)
Assertion
Ref Expression
isinftm ((𝑊𝑉𝑋𝐵𝑌𝐵) → (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.
StepHypRef Expression
1 eleq1 2676 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐵𝑋𝐵))
2 eleq1 2676 . . . . . 6 (𝑦 = 𝑌 → (𝑦𝐵𝑌𝐵))
31, 2bi2anan9 913 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥𝐵𝑦𝐵) ↔ (𝑋𝐵𝑌𝐵)))
4 simpl 472 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
54breq2d 4595 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ( 0 < 𝑥0 < 𝑋))
64oveq2d 6565 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑛 · 𝑥) = (𝑛 · 𝑋))
7 simpr 476 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
86, 7breq12d 4596 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑛 · 𝑥) < 𝑦 ↔ (𝑛 · 𝑋) < 𝑌))
98ralbidv 2969 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))
105, 9anbi12d 743 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦) ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
113, 10anbi12d 743 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦)) ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
12 eqid 2610 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}
1311, 12brabga 4914 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
14133adant1 1072 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
15 elex 3185 . . . . 5 (𝑊𝑉𝑊 ∈ V)
16153ad2ant1 1075 . . . 4 ((𝑊𝑉𝑋𝐵𝑌𝐵) → 𝑊 ∈ V)
17 fveq2 6103 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
18 inftm.b . . . . . . . . . 10 𝐵 = (Base‘𝑊)
1917, 18syl6eqr 2662 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
2019eleq2d 2673 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥𝐵))
2119eleq2d 2673 . . . . . . . 8 (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦𝐵))
2220, 21anbi12d 743 . . . . . . 7 (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥𝐵𝑦𝐵)))
23 fveq2 6103 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
24 inftm.0 . . . . . . . . . 10 0 = (0g𝑊)
2523, 24syl6eqr 2662 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = 0 )
26 fveq2 6103 . . . . . . . . . 10 (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
27 inftm.l . . . . . . . . . 10 < = (lt‘𝑊)
2826, 27syl6eqr 2662 . . . . . . . . 9 (𝑤 = 𝑊 → (lt‘𝑤) = < )
29 eqidd 2611 . . . . . . . . 9 (𝑤 = 𝑊𝑥 = 𝑥)
3025, 28, 29breq123d 4597 . . . . . . . 8 (𝑤 = 𝑊 → ((0g𝑤)(lt‘𝑤)𝑥0 < 𝑥))
31 fveq2 6103 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (.g𝑤) = (.g𝑊))
32 inftm.x . . . . . . . . . . . 12 · = (.g𝑊)
3331, 32syl6eqr 2662 . . . . . . . . . . 11 (𝑤 = 𝑊 → (.g𝑤) = · )
3433oveqd 6566 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑛(.g𝑤)𝑥) = (𝑛 · 𝑥))
35 eqidd 2611 . . . . . . . . . 10 (𝑤 = 𝑊𝑦 = 𝑦)
3634, 28, 35breq123d 4597 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛 · 𝑥) < 𝑦))
3736ralbidv 2969 . . . . . . . 8 (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))
3830, 37anbi12d 743 . . . . . . 7 (𝑤 = 𝑊 → (((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦) ↔ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦)))
3922, 38anbi12d 743 . . . . . 6 (𝑤 = 𝑊 → (((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))))
4039opabbidv 4648 . . . . 5 (𝑤 = 𝑊 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))})
41 df-inftm 29063 . . . . 5 ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
42 fvex 6113 . . . . . . . 8 (Base‘𝑊) ∈ V
4318, 42eqeltri 2684 . . . . . . 7 𝐵 ∈ V
4443, 43xpex 6860 . . . . . 6 (𝐵 × 𝐵) ∈ V
45 opabssxp 5116 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ⊆ (𝐵 × 𝐵)
4644, 45ssexi 4731 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ∈ V
4740, 41, 46fvmpt 6191 . . . 4 (𝑊 ∈ V → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))})
4816, 47syl 17 . . 3 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))})
4948breqd 4594 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌))
50 3simpc 1053 . . 3 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋𝐵𝑌𝐵))
5150biantrurd 528 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌) ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
5214, 49, 513bitr4d 299 1 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   class class class wbr 4583  {copab 4642   × cxp 5036  ‘cfv 5804  (class class class)co 6549  ℕcn 10897  Basecbs 15695  0gc0g 15923  ltcplt 16764  .gcmg 17363  ⋘cinftm 29061 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-inftm 29063 This theorem is referenced by:  pnfinf  29068  isarchi2  29070
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