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Theorem fiunelros 30010
Description: A ring of sets is closed under finite union. (Contributed by Thierry Arnoux, 19-Jul-2020.)
Hypotheses
Ref Expression
isros.1 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
fiunelros.1 (𝜑𝑆𝑄)
fiunelros.2 (𝜑𝑁 ∈ ℕ)
fiunelros.3 ((𝜑𝑘 ∈ (1..^𝑁)) → 𝐵𝑆)
Assertion
Ref Expression
fiunelros (𝜑 𝑘 ∈ (1..^𝑁)𝐵𝑆)
Distinct variable groups:   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦   𝑘,𝑁   𝑆,𝑘   𝜑,𝑘
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑠)   𝐵(𝑥,𝑦,𝑘,𝑠)   𝑄(𝑥,𝑦,𝑘,𝑠)   𝑁(𝑥,𝑦,𝑠)   𝑂(𝑥,𝑦,𝑘)

Proof of Theorem fiunelros
Dummy variables 𝑖 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fiunelros.2 . 2 (𝜑𝑁 ∈ ℕ)
2 simpr 477 . . . . 5 ((𝜑𝑁 ∈ ℕ) → 𝑁 ∈ ℕ)
32nnred 10980 . . . 4 ((𝜑𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
43leidd 10539 . . 3 ((𝜑𝑁 ∈ ℕ) → 𝑁𝑁)
5 breq1 4621 . . . . 5 (𝑛 = 1 → (𝑛𝑁 ↔ 1 ≤ 𝑁))
6 oveq2 6613 . . . . . . 7 (𝑛 = 1 → (1..^𝑛) = (1..^1))
76iuneq1d 4516 . . . . . 6 (𝑛 = 1 → 𝑘 ∈ (1..^𝑛)𝐵 = 𝑘 ∈ (1..^1)𝐵)
87eleq1d 2688 . . . . 5 (𝑛 = 1 → ( 𝑘 ∈ (1..^𝑛)𝐵𝑆 𝑘 ∈ (1..^1)𝐵𝑆))
95, 8imbi12d 334 . . . 4 (𝑛 = 1 → ((𝑛𝑁 𝑘 ∈ (1..^𝑛)𝐵𝑆) ↔ (1 ≤ 𝑁 𝑘 ∈ (1..^1)𝐵𝑆)))
10 breq1 4621 . . . . 5 (𝑛 = 𝑖 → (𝑛𝑁𝑖𝑁))
11 oveq2 6613 . . . . . . 7 (𝑛 = 𝑖 → (1..^𝑛) = (1..^𝑖))
1211iuneq1d 4516 . . . . . 6 (𝑛 = 𝑖 𝑘 ∈ (1..^𝑛)𝐵 = 𝑘 ∈ (1..^𝑖)𝐵)
1312eleq1d 2688 . . . . 5 (𝑛 = 𝑖 → ( 𝑘 ∈ (1..^𝑛)𝐵𝑆 𝑘 ∈ (1..^𝑖)𝐵𝑆))
1410, 13imbi12d 334 . . . 4 (𝑛 = 𝑖 → ((𝑛𝑁 𝑘 ∈ (1..^𝑛)𝐵𝑆) ↔ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)))
15 breq1 4621 . . . . 5 (𝑛 = (𝑖 + 1) → (𝑛𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
16 oveq2 6613 . . . . . . 7 (𝑛 = (𝑖 + 1) → (1..^𝑛) = (1..^(𝑖 + 1)))
1716iuneq1d 4516 . . . . . 6 (𝑛 = (𝑖 + 1) → 𝑘 ∈ (1..^𝑛)𝐵 = 𝑘 ∈ (1..^(𝑖 + 1))𝐵)
1817eleq1d 2688 . . . . 5 (𝑛 = (𝑖 + 1) → ( 𝑘 ∈ (1..^𝑛)𝐵𝑆 𝑘 ∈ (1..^(𝑖 + 1))𝐵𝑆))
1915, 18imbi12d 334 . . . 4 (𝑛 = (𝑖 + 1) → ((𝑛𝑁 𝑘 ∈ (1..^𝑛)𝐵𝑆) ↔ ((𝑖 + 1) ≤ 𝑁 𝑘 ∈ (1..^(𝑖 + 1))𝐵𝑆)))
20 breq1 4621 . . . . 5 (𝑛 = 𝑁 → (𝑛𝑁𝑁𝑁))
21 oveq2 6613 . . . . . . 7 (𝑛 = 𝑁 → (1..^𝑛) = (1..^𝑁))
2221iuneq1d 4516 . . . . . 6 (𝑛 = 𝑁 𝑘 ∈ (1..^𝑛)𝐵 = 𝑘 ∈ (1..^𝑁)𝐵)
2322eleq1d 2688 . . . . 5 (𝑛 = 𝑁 → ( 𝑘 ∈ (1..^𝑛)𝐵𝑆 𝑘 ∈ (1..^𝑁)𝐵𝑆))
2420, 23imbi12d 334 . . . 4 (𝑛 = 𝑁 → ((𝑛𝑁 𝑘 ∈ (1..^𝑛)𝐵𝑆) ↔ (𝑁𝑁 𝑘 ∈ (1..^𝑁)𝐵𝑆)))
25 fzo0 12430 . . . . . . . 8 (1..^1) = ∅
26 iuneq1 4505 . . . . . . . 8 ((1..^1) = ∅ → 𝑘 ∈ (1..^1)𝐵 = 𝑘 ∈ ∅ 𝐵)
2725, 26ax-mp 5 . . . . . . 7 𝑘 ∈ (1..^1)𝐵 = 𝑘 ∈ ∅ 𝐵
28 0iun 4548 . . . . . . 7 𝑘 ∈ ∅ 𝐵 = ∅
2927, 28eqtri 2648 . . . . . 6 𝑘 ∈ (1..^1)𝐵 = ∅
30 fiunelros.1 . . . . . . 7 (𝜑𝑆𝑄)
31 isros.1 . . . . . . . 8 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
32310elros 30006 . . . . . . 7 (𝑆𝑄 → ∅ ∈ 𝑆)
3330, 32syl 17 . . . . . 6 (𝜑 → ∅ ∈ 𝑆)
3429, 33syl5eqel 2708 . . . . 5 (𝜑 𝑘 ∈ (1..^1)𝐵𝑆)
3534a1d 25 . . . 4 (𝜑 → (1 ≤ 𝑁 𝑘 ∈ (1..^1)𝐵𝑆))
36 simpllr 798 . . . . . . . 8 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ ℕ)
37 fzosplitsn 12514 . . . . . . . . . 10 (𝑖 ∈ (ℤ‘1) → (1..^(𝑖 + 1)) = ((1..^𝑖) ∪ {𝑖}))
38 nnuz 11667 . . . . . . . . . 10 ℕ = (ℤ‘1)
3937, 38eleq2s 2722 . . . . . . . . 9 (𝑖 ∈ ℕ → (1..^(𝑖 + 1)) = ((1..^𝑖) ∪ {𝑖}))
4039iuneq1d 4516 . . . . . . . 8 (𝑖 ∈ ℕ → 𝑘 ∈ (1..^(𝑖 + 1))𝐵 = 𝑘 ∈ ((1..^𝑖) ∪ {𝑖})𝐵)
4136, 40syl 17 . . . . . . 7 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑘 ∈ (1..^(𝑖 + 1))𝐵 = 𝑘 ∈ ((1..^𝑖) ∪ {𝑖})𝐵)
42 iunxun 4576 . . . . . . 7 𝑘 ∈ ((1..^𝑖) ∪ {𝑖})𝐵 = ( 𝑘 ∈ (1..^𝑖)𝐵 𝑘 ∈ {𝑖}𝐵)
4341, 42syl6eq 2676 . . . . . 6 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑘 ∈ (1..^(𝑖 + 1))𝐵 = ( 𝑘 ∈ (1..^𝑖)𝐵 𝑘 ∈ {𝑖}𝐵))
4430ad3antrrr 765 . . . . . . 7 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑆𝑄)
4536nnred 10980 . . . . . . . . 9 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ ℝ)
461ad3antrrr 765 . . . . . . . . . 10 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑁 ∈ ℕ)
4746nnred 10980 . . . . . . . . 9 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑁 ∈ ℝ)
48 simpr 477 . . . . . . . . . 10 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 + 1) ≤ 𝑁)
49 nnltp1le 11378 . . . . . . . . . . 11 ((𝑖 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑖 < 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
5036, 46, 49syl2anc 692 . . . . . . . . . 10 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖 < 𝑁 ↔ (𝑖 + 1) ≤ 𝑁))
5148, 50mpbird 247 . . . . . . . . 9 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 < 𝑁)
5245, 47, 51ltled 10130 . . . . . . . 8 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖𝑁)
53 simplr 791 . . . . . . . 8 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆))
5452, 53mpd 15 . . . . . . 7 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑘 ∈ (1..^𝑖)𝐵𝑆)
55 nfcsb1v 3535 . . . . . . . . . 10 𝑘𝑖 / 𝑘𝐵
56 csbeq1a 3528 . . . . . . . . . 10 (𝑘 = 𝑖𝐵 = 𝑖 / 𝑘𝐵)
5755, 56iunxsngf 29212 . . . . . . . . 9 (𝑖 ∈ ℕ → 𝑘 ∈ {𝑖}𝐵 = 𝑖 / 𝑘𝐵)
5836, 57syl 17 . . . . . . . 8 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑘 ∈ {𝑖}𝐵 = 𝑖 / 𝑘𝐵)
59 simplll 797 . . . . . . . . 9 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝜑)
60 elfzo1 12455 . . . . . . . . . 10 (𝑖 ∈ (1..^𝑁) ↔ (𝑖 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑖 < 𝑁))
6136, 46, 51, 60syl3anbrc 1244 . . . . . . . . 9 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 ∈ (1..^𝑁))
62 nfv 1845 . . . . . . . . . . 11 𝑘(𝜑𝑖 ∈ (1..^𝑁))
63 nfcv 2767 . . . . . . . . . . . 12 𝑘𝑆
6455, 63nfel 2779 . . . . . . . . . . 11 𝑘𝑖 / 𝑘𝐵𝑆
6562, 64nfim 1827 . . . . . . . . . 10 𝑘((𝜑𝑖 ∈ (1..^𝑁)) → 𝑖 / 𝑘𝐵𝑆)
66 eleq1 2692 . . . . . . . . . . . 12 (𝑘 = 𝑖 → (𝑘 ∈ (1..^𝑁) ↔ 𝑖 ∈ (1..^𝑁)))
6766anbi2d 739 . . . . . . . . . . 11 (𝑘 = 𝑖 → ((𝜑𝑘 ∈ (1..^𝑁)) ↔ (𝜑𝑖 ∈ (1..^𝑁))))
6856eleq1d 2688 . . . . . . . . . . 11 (𝑘 = 𝑖 → (𝐵𝑆𝑖 / 𝑘𝐵𝑆))
6967, 68imbi12d 334 . . . . . . . . . 10 (𝑘 = 𝑖 → (((𝜑𝑘 ∈ (1..^𝑁)) → 𝐵𝑆) ↔ ((𝜑𝑖 ∈ (1..^𝑁)) → 𝑖 / 𝑘𝐵𝑆)))
70 fiunelros.3 . . . . . . . . . 10 ((𝜑𝑘 ∈ (1..^𝑁)) → 𝐵𝑆)
7165, 69, 70chvar 2266 . . . . . . . . 9 ((𝜑𝑖 ∈ (1..^𝑁)) → 𝑖 / 𝑘𝐵𝑆)
7259, 61, 71syl2anc 692 . . . . . . . 8 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑖 / 𝑘𝐵𝑆)
7358, 72eqeltrd 2704 . . . . . . 7 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑘 ∈ {𝑖}𝐵𝑆)
7431unelros 30007 . . . . . . 7 ((𝑆𝑄 𝑘 ∈ (1..^𝑖)𝐵𝑆 𝑘 ∈ {𝑖}𝐵𝑆) → ( 𝑘 ∈ (1..^𝑖)𝐵 𝑘 ∈ {𝑖}𝐵) ∈ 𝑆)
7544, 54, 73, 74syl3anc 1323 . . . . . 6 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → ( 𝑘 ∈ (1..^𝑖)𝐵 𝑘 ∈ {𝑖}𝐵) ∈ 𝑆)
7643, 75eqeltrd 2704 . . . . 5 ((((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) ∧ (𝑖 + 1) ≤ 𝑁) → 𝑘 ∈ (1..^(𝑖 + 1))𝐵𝑆)
7776ex 450 . . . 4 (((𝜑𝑖 ∈ ℕ) ∧ (𝑖𝑁 𝑘 ∈ (1..^𝑖)𝐵𝑆)) → ((𝑖 + 1) ≤ 𝑁 𝑘 ∈ (1..^(𝑖 + 1))𝐵𝑆))
789, 14, 19, 24, 35, 77nnindd 29399 . . 3 ((𝜑𝑁 ∈ ℕ) → (𝑁𝑁 𝑘 ∈ (1..^𝑁)𝐵𝑆))
794, 78mpd 15 . 2 ((𝜑𝑁 ∈ ℕ) → 𝑘 ∈ (1..^𝑁)𝐵𝑆)
801, 79mpdan 701 1 (𝜑 𝑘 ∈ (1..^𝑁)𝐵𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  {crab 2916  csb 3519  cdif 3557  cun 3558  c0 3896  𝒫 cpw 4135  {csn 4153   ciun 4490   class class class wbr 4618  cfv 5850  (class class class)co 6605  1c1 9882   + caddc 9884   < clt 10019  cle 10020  cn 10965  cuz 11631  ..^cfzo 12403
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404
This theorem is referenced by: (None)
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