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Theorem isinftm 33171
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 2827 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐵𝑋𝐵))
2 eleq1 2827 . . . . . 6 (𝑦 = 𝑌 → (𝑦𝐵𝑌𝐵))
31, 2bi2anan9 638 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥𝐵𝑦𝐵) ↔ (𝑋𝐵𝑌𝐵)))
4 simpl 482 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
54breq2d 5160 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ( 0 < 𝑥0 < 𝑋))
64oveq2d 7447 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑛 · 𝑥) = (𝑛 · 𝑋))
7 simpr 484 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
86, 7breq12d 5161 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑛 · 𝑥) < 𝑦 ↔ (𝑛 · 𝑋) < 𝑌))
98ralbidv 3176 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))
105, 9anbi12d 632 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦) ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
113, 10anbi12d 632 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦)) ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
12 eqid 2735 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}
1311, 12brabga 5544 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
14133adant1 1129 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
15 elex 3499 . . . . 5 (𝑊𝑉𝑊 ∈ V)
16153ad2ant1 1132 . . . 4 ((𝑊𝑉𝑋𝐵𝑌𝐵) → 𝑊 ∈ V)
17 fveq2 6907 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
18 inftm.b . . . . . . . . . 10 𝐵 = (Base‘𝑊)
1917, 18eqtr4di 2793 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
2019eleq2d 2825 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥𝐵))
2119eleq2d 2825 . . . . . . . 8 (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦𝐵))
2220, 21anbi12d 632 . . . . . . 7 (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥𝐵𝑦𝐵)))
23 fveq2 6907 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
24 inftm.0 . . . . . . . . . 10 0 = (0g𝑊)
2523, 24eqtr4di 2793 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = 0 )
26 fveq2 6907 . . . . . . . . . 10 (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
27 inftm.l . . . . . . . . . 10 < = (lt‘𝑊)
2826, 27eqtr4di 2793 . . . . . . . . 9 (𝑤 = 𝑊 → (lt‘𝑤) = < )
29 eqidd 2736 . . . . . . . . 9 (𝑤 = 𝑊𝑥 = 𝑥)
3025, 28, 29breq123d 5162 . . . . . . . 8 (𝑤 = 𝑊 → ((0g𝑤)(lt‘𝑤)𝑥0 < 𝑥))
31 fveq2 6907 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (.g𝑤) = (.g𝑊))
32 inftm.x . . . . . . . . . . . 12 · = (.g𝑊)
3331, 32eqtr4di 2793 . . . . . . . . . . 11 (𝑤 = 𝑊 → (.g𝑤) = · )
3433oveqd 7448 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑛(.g𝑤)𝑥) = (𝑛 · 𝑥))
35 eqidd 2736 . . . . . . . . . 10 (𝑤 = 𝑊𝑦 = 𝑦)
3634, 28, 35breq123d 5162 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛 · 𝑥) < 𝑦))
3736ralbidv 3176 . . . . . . . 8 (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))
3830, 37anbi12d 632 . . . . . . 7 (𝑤 = 𝑊 → (((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦) ↔ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦)))
3922, 38anbi12d 632 . . . . . 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 5214 . . . . 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 33168 . . . . 5 ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
4218fvexi 6921 . . . . . . 7 𝐵 ∈ V
4342, 42xpex 7772 . . . . . 6 (𝐵 × 𝐵) ∈ V
44 opabssxp 5781 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ⊆ (𝐵 × 𝐵)
4543, 44ssexi 5328 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ∈ V
4640, 41, 45fvmpt 7016 . . . 4 (𝑊 ∈ V → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))})
4716, 46syl 17 . . 3 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (⋘‘𝑊) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))})
4847breqd 5159 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌))
49 3simpc 1149 . . 3 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋𝐵𝑌𝐵))
5049biantrurd 532 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌) ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
5114, 48, 503bitr4d 311 1 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478   class class class wbr 5148  {copab 5210   × cxp 5687  cfv 6563  (class class class)co 7431  cn 12264  Basecbs 17245  0gc0g 17486  ltcplt 18366  .gcmg 19098  cinftm 33166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-inftm 33168
This theorem is referenced by:  pnfinf  33173  isarchi2  33175
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