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Theorem isinftm 29517
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 2686 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐵𝑋𝐵))
2 eleq1 2686 . . . . . 6 (𝑦 = 𝑌 → (𝑦𝐵𝑌𝐵))
31, 2bi2anan9 916 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥𝐵𝑦𝐵) ↔ (𝑋𝐵𝑌𝐵)))
4 simpl 473 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
54breq2d 4625 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ( 0 < 𝑥0 < 𝑋))
64oveq2d 6620 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑛 · 𝑥) = (𝑛 · 𝑋))
7 simpr 477 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
86, 7breq12d 4626 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑛 · 𝑥) < 𝑦 ↔ (𝑛 · 𝑋) < 𝑌))
98ralbidv 2980 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))
105, 9anbi12d 746 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦) ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
113, 10anbi12d 746 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦)) ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
12 eqid 2621 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}
1311, 12brabga 4949 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
14133adant1 1077 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
15 elex 3198 . . . . 5 (𝑊𝑉𝑊 ∈ V)
16153ad2ant1 1080 . . . 4 ((𝑊𝑉𝑋𝐵𝑌𝐵) → 𝑊 ∈ V)
17 fveq2 6148 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
18 inftm.b . . . . . . . . . 10 𝐵 = (Base‘𝑊)
1917, 18syl6eqr 2673 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
2019eleq2d 2684 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥𝐵))
2119eleq2d 2684 . . . . . . . 8 (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦𝐵))
2220, 21anbi12d 746 . . . . . . 7 (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥𝐵𝑦𝐵)))
23 fveq2 6148 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
24 inftm.0 . . . . . . . . . 10 0 = (0g𝑊)
2523, 24syl6eqr 2673 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = 0 )
26 fveq2 6148 . . . . . . . . . 10 (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
27 inftm.l . . . . . . . . . 10 < = (lt‘𝑊)
2826, 27syl6eqr 2673 . . . . . . . . 9 (𝑤 = 𝑊 → (lt‘𝑤) = < )
29 eqidd 2622 . . . . . . . . 9 (𝑤 = 𝑊𝑥 = 𝑥)
3025, 28, 29breq123d 4627 . . . . . . . 8 (𝑤 = 𝑊 → ((0g𝑤)(lt‘𝑤)𝑥0 < 𝑥))
31 fveq2 6148 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (.g𝑤) = (.g𝑊))
32 inftm.x . . . . . . . . . . . 12 · = (.g𝑊)
3331, 32syl6eqr 2673 . . . . . . . . . . 11 (𝑤 = 𝑊 → (.g𝑤) = · )
3433oveqd 6621 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑛(.g𝑤)𝑥) = (𝑛 · 𝑥))
35 eqidd 2622 . . . . . . . . . 10 (𝑤 = 𝑊𝑦 = 𝑦)
3634, 28, 35breq123d 4627 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛 · 𝑥) < 𝑦))
3736ralbidv 2980 . . . . . . . 8 (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))
3830, 37anbi12d 746 . . . . . . 7 (𝑤 = 𝑊 → (((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦) ↔ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦)))
3922, 38anbi12d 746 . . . . . 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 4678 . . . . 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 29514 . . . . 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 6158 . . . . . . . 8 (Base‘𝑊) ∈ V
4318, 42eqeltri 2694 . . . . . . 7 𝐵 ∈ V
4443, 43xpex 6915 . . . . . 6 (𝐵 × 𝐵) ∈ V
45 opabssxp 5154 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ⊆ (𝐵 × 𝐵)
4644, 45ssexi 4763 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ∈ V
4740, 41, 46fvmpt 6239 . . . 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 4624 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌))
50 3simpc 1058 . . 3 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋𝐵𝑌𝐵))
5150biantrurd 529 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌) ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
5214, 49, 513bitr4d 300 1 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186   class class class wbr 4613  {copab 4672   × cxp 5072  cfv 5847  (class class class)co 6604  cn 10964  Basecbs 15781  0gc0g 16021  ltcplt 16862  .gcmg 17461  cinftm 29512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-inftm 29514
This theorem is referenced by:  pnfinf  29519  isarchi2  29521
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