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Theorem isinftm 30867
 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 2877 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐵𝑋𝐵))
2 eleq1 2877 . . . . . 6 (𝑦 = 𝑌 → (𝑦𝐵𝑌𝐵))
31, 2bi2anan9 638 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑥𝐵𝑦𝐵) ↔ (𝑋𝐵𝑌𝐵)))
4 simpl 486 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑥 = 𝑋)
54breq2d 5042 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → ( 0 < 𝑥0 < 𝑋))
64oveq2d 7151 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑛 · 𝑥) = (𝑛 · 𝑋))
7 simpr 488 . . . . . . . 8 ((𝑥 = 𝑋𝑦 = 𝑌) → 𝑦 = 𝑌)
86, 7breq12d 5043 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → ((𝑛 · 𝑥) < 𝑦 ↔ (𝑛 · 𝑋) < 𝑌))
98ralbidv 3162 . . . . . 6 ((𝑥 = 𝑋𝑦 = 𝑌) → (∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))
105, 9anbi12d 633 . . . . 5 ((𝑥 = 𝑋𝑦 = 𝑌) → (( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦) ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
113, 10anbi12d 633 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦)) ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
12 eqid 2798 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}
1311, 12brabga 5386 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
14133adant1 1127 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
15 elex 3459 . . . . 5 (𝑊𝑉𝑊 ∈ V)
16153ad2ant1 1130 . . . 4 ((𝑊𝑉𝑋𝐵𝑌𝐵) → 𝑊 ∈ V)
17 fveq2 6645 . . . . . . . . . 10 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
18 inftm.b . . . . . . . . . 10 𝐵 = (Base‘𝑊)
1917, 18eqtr4di 2851 . . . . . . . . 9 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
2019eleq2d 2875 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↔ 𝑥𝐵))
2119eleq2d 2875 . . . . . . . 8 (𝑤 = 𝑊 → (𝑦 ∈ (Base‘𝑤) ↔ 𝑦𝐵))
2220, 21anbi12d 633 . . . . . . 7 (𝑤 = 𝑊 → ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ↔ (𝑥𝐵𝑦𝐵)))
23 fveq2 6645 . . . . . . . . . 10 (𝑤 = 𝑊 → (0g𝑤) = (0g𝑊))
24 inftm.0 . . . . . . . . . 10 0 = (0g𝑊)
2523, 24eqtr4di 2851 . . . . . . . . 9 (𝑤 = 𝑊 → (0g𝑤) = 0 )
26 fveq2 6645 . . . . . . . . . 10 (𝑤 = 𝑊 → (lt‘𝑤) = (lt‘𝑊))
27 inftm.l . . . . . . . . . 10 < = (lt‘𝑊)
2826, 27eqtr4di 2851 . . . . . . . . 9 (𝑤 = 𝑊 → (lt‘𝑤) = < )
29 eqidd 2799 . . . . . . . . 9 (𝑤 = 𝑊𝑥 = 𝑥)
3025, 28, 29breq123d 5044 . . . . . . . 8 (𝑤 = 𝑊 → ((0g𝑤)(lt‘𝑤)𝑥0 < 𝑥))
31 fveq2 6645 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (.g𝑤) = (.g𝑊))
32 inftm.x . . . . . . . . . . . 12 · = (.g𝑊)
3331, 32eqtr4di 2851 . . . . . . . . . . 11 (𝑤 = 𝑊 → (.g𝑤) = · )
3433oveqd 7152 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑛(.g𝑤)𝑥) = (𝑛 · 𝑥))
35 eqidd 2799 . . . . . . . . . 10 (𝑤 = 𝑊𝑦 = 𝑦)
3634, 28, 35breq123d 5044 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ (𝑛 · 𝑥) < 𝑦))
3736ralbidv 3162 . . . . . . . 8 (𝑤 = 𝑊 → (∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦 ↔ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))
3830, 37anbi12d 633 . . . . . . 7 (𝑤 = 𝑊 → (((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦) ↔ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦)))
3922, 38anbi12d 633 . . . . . 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 5096 . . . . 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 30864 . . . . 5 ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
4218fvexi 6659 . . . . . . 7 𝐵 ∈ V
4342, 42xpex 7458 . . . . . 6 (𝐵 × 𝐵) ∈ V
44 opabssxp 5607 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ⊆ (𝐵 × 𝐵)
4543, 44ssexi 5190 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))} ∈ V
4640, 41, 45fvmpt 6745 . . . 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 5041 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌𝑋{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ( 0 < 𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑥) < 𝑦))}𝑌))
49 3simpc 1147 . . 3 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋𝐵𝑌𝐵))
5049biantrurd 536 . 2 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌) ↔ ((𝑋𝐵𝑌𝐵) ∧ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))))
5114, 48, 503bitr4d 314 1 ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   class class class wbr 5030  {copab 5092   × cxp 5517  ‘cfv 6324  (class class class)co 7135  ℕcn 11627  Basecbs 16477  0gc0g 16707  ltcplt 17545  .gcmg 18219  ⋘cinftm 30862 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-inftm 30864 This theorem is referenced by:  pnfinf  30869  isarchi2  30871
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