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Theorem iuneqfzuzlem 39863
Description: Lemma for iuneqfzuz 39864: here, inclusion is proven; aiuneqfzuz uses this lemma twice, to prove equality. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
iuneqfzuzlem.z 𝑍 = (ℤ𝑁)
Assertion
Ref Expression
iuneqfzuzlem (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐴 𝑛𝑍 𝐵)
Distinct variable groups:   𝐴,𝑚   𝐵,𝑚   𝑛,𝑁   𝑚,𝑍,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝐵(𝑛)   𝑁(𝑚)

Proof of Theorem iuneqfzuzlem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2793 . . . . . . . . 9 𝑚𝐴
2 nfcsb1v 3582 . . . . . . . . 9 𝑛𝑚 / 𝑛𝐴
3 csbeq1a 3575 . . . . . . . . 9 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
41, 2, 3cbviun 4589 . . . . . . . 8 𝑛𝑍 𝐴 = 𝑚𝑍 𝑚 / 𝑛𝐴
54eleq2i 2722 . . . . . . 7 (𝑥 𝑛𝑍 𝐴𝑥 𝑚𝑍 𝑚 / 𝑛𝐴)
6 eliun 4556 . . . . . . 7 (𝑥 𝑚𝑍 𝑚 / 𝑛𝐴 ↔ ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
75, 6bitri 264 . . . . . 6 (𝑥 𝑛𝑍 𝐴 ↔ ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
87biimpi 206 . . . . 5 (𝑥 𝑛𝑍 𝐴 → ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
98adantl 481 . . . 4 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥 𝑛𝑍 𝐴) → ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
10 nfra1 2970 . . . . . 6 𝑚𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵
11 nfv 1883 . . . . . 6 𝑚 𝑥 𝑛𝑍 𝐵
12 simp2 1082 . . . . . . . 8 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑚𝑍)
13 rspa 2959 . . . . . . . . 9 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍) → 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵)
14133adant3 1101 . . . . . . . 8 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵)
15 simp3 1083 . . . . . . . 8 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑥𝑚 / 𝑛𝐴)
16 id 22 . . . . . . . . . . 11 ( 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵)
17 fzssuz 12420 . . . . . . . . . . . . 13 (𝑁...𝑚) ⊆ (ℤ𝑁)
18 iuneqfzuzlem.z . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑁)
1918eqcomi 2660 . . . . . . . . . . . . 13 (ℤ𝑁) = 𝑍
2017, 19sseqtri 3670 . . . . . . . . . . . 12 (𝑁...𝑚) ⊆ 𝑍
21 iunss1 4564 . . . . . . . . . . . 12 ((𝑁...𝑚) ⊆ 𝑍 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐵)
2220, 21mp1i 13 . . . . . . . . . . 11 ( 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐵)
2316, 22eqsstrd 3672 . . . . . . . . . 10 ( 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛 ∈ (𝑁...𝑚)𝐴 𝑛𝑍 𝐵)
24233ad2ant2 1103 . . . . . . . . 9 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑛 ∈ (𝑁...𝑚)𝐴 𝑛𝑍 𝐵)
2518eleq2i 2722 . . . . . . . . . . . . . . . . 17 (𝑚𝑍𝑚 ∈ (ℤ𝑁))
2625biimpi 206 . . . . . . . . . . . . . . . 16 (𝑚𝑍𝑚 ∈ (ℤ𝑁))
27 eluzel2 11730 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (ℤ𝑁) → 𝑁 ∈ ℤ)
2826, 27syl 17 . . . . . . . . . . . . . . 15 (𝑚𝑍𝑁 ∈ ℤ)
29 eluzelz 11735 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (ℤ𝑁) → 𝑚 ∈ ℤ)
3026, 29syl 17 . . . . . . . . . . . . . . 15 (𝑚𝑍𝑚 ∈ ℤ)
3128, 30, 303jca 1261 . . . . . . . . . . . . . 14 (𝑚𝑍 → (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ))
32 eluzle 11738 . . . . . . . . . . . . . . 15 (𝑚 ∈ (ℤ𝑁) → 𝑁𝑚)
3326, 32syl 17 . . . . . . . . . . . . . 14 (𝑚𝑍𝑁𝑚)
3430zred 11520 . . . . . . . . . . . . . . 15 (𝑚𝑍𝑚 ∈ ℝ)
35 leid 10171 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℝ → 𝑚𝑚)
3634, 35syl 17 . . . . . . . . . . . . . 14 (𝑚𝑍𝑚𝑚)
3731, 33, 36jca32 557 . . . . . . . . . . . . 13 (𝑚𝑍 → ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑁𝑚𝑚𝑚)))
38 elfz2 12371 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑁...𝑚) ↔ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑁𝑚𝑚𝑚)))
3937, 38sylibr 224 . . . . . . . . . . . 12 (𝑚𝑍𝑚 ∈ (𝑁...𝑚))
40 nfcv 2793 . . . . . . . . . . . . . 14 𝑛𝑥
4140, 2nfel 2806 . . . . . . . . . . . . 13 𝑛 𝑥𝑚 / 𝑛𝐴
423eleq2d 2716 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝑥𝐴𝑥𝑚 / 𝑛𝐴))
4341, 42rspce 3335 . . . . . . . . . . . 12 ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥𝑚 / 𝑛𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
4439, 43sylan 487 . . . . . . . . . . 11 ((𝑚𝑍𝑥𝑚 / 𝑛𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
45 eliun 4556 . . . . . . . . . . 11 (𝑥 𝑛 ∈ (𝑁...𝑚)𝐴 ↔ ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
4644, 45sylibr 224 . . . . . . . . . 10 ((𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛 ∈ (𝑁...𝑚)𝐴)
47463adant2 1100 . . . . . . . . 9 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛 ∈ (𝑁...𝑚)𝐴)
4824, 47sseldd 3637 . . . . . . . 8 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛𝑍 𝐵)
4912, 14, 15, 48syl3anc 1366 . . . . . . 7 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛𝑍 𝐵)
50493exp 1283 . . . . . 6 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → (𝑚𝑍 → (𝑥𝑚 / 𝑛𝐴𝑥 𝑛𝑍 𝐵)))
5110, 11, 50rexlimd 3055 . . . . 5 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → (∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴𝑥 𝑛𝑍 𝐵))
5251adantr 480 . . . 4 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥 𝑛𝑍 𝐴) → (∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴𝑥 𝑛𝑍 𝐵))
539, 52mpd 15 . . 3 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥 𝑛𝑍 𝐴) → 𝑥 𝑛𝑍 𝐵)
5453ralrimiva 2995 . 2 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑥 𝑛𝑍 𝐴𝑥 𝑛𝑍 𝐵)
55 dfss3 3625 . 2 ( 𝑛𝑍 𝐴 𝑛𝑍 𝐵 ↔ ∀𝑥 𝑛𝑍 𝐴𝑥 𝑛𝑍 𝐵)
5654, 55sylibr 224 1 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛𝑍 𝐴 𝑛𝑍 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  csb 3566  wss 3607   ciun 4552   class class class wbr 4685  cfv 5926  (class class class)co 6690  cr 9973  cle 10113  cz 11415  cuz 11725  ...cfz 12364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-neg 10307  df-z 11416  df-uz 11726  df-fz 12365
This theorem is referenced by:  iuneqfzuz  39864
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