Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iuneqfzuzlem Structured version   Visualization version   GIF version

Theorem iuneqfzuzlem 39011
Description: Lemma for iuneqfzuz 39012: 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   𝑛,𝑍,𝑚
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 2761 . . . . . . . . 9 𝑚𝐴
2 nfcsb1v 3530 . . . . . . . . 9 𝑛𝑚 / 𝑛𝐴
3 csbeq1a 3523 . . . . . . . . 9 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
41, 2, 3cbviun 4523 . . . . . . . 8 𝑛𝑍 𝐴 = 𝑚𝑍 𝑚 / 𝑛𝐴
54eleq2i 2690 . . . . . . 7 (𝑥 𝑛𝑍 𝐴𝑥 𝑚𝑍 𝑚 / 𝑛𝐴)
6 eliun 4490 . . . . . . 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 482 . . . 4 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥 𝑛𝑍 𝐴) → ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
10 nfra1 2936 . . . . . 6 𝑚𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵
11 nfv 1840 . . . . . 6 𝑚 𝑥 𝑛𝑍 𝐵
12 simp2 1060 . . . . . . . 8 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑚𝑍)
13 rspa 2925 . . . . . . . . 9 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍) → 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵)
14133adant3 1079 . . . . . . . 8 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵)
15 simp3 1061 . . . . . . . 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 12324 . . . . . . . . . . . . 13 (𝑁...𝑚) ⊆ (ℤ𝑁)
18 iuneqfzuzlem.z . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑁)
1918eqcomi 2630 . . . . . . . . . . . . 13 (ℤ𝑁) = 𝑍
2017, 19sseqtri 3616 . . . . . . . . . . . 12 (𝑁...𝑚) ⊆ 𝑍
21 iunss1 4498 . . . . . . . . . . . 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 3618 . . . . . . . . . 10 ( 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛 ∈ (𝑁...𝑚)𝐴 𝑛𝑍 𝐵)
24233ad2ant2 1081 . . . . . . . . 9 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑛 ∈ (𝑁...𝑚)𝐴 𝑛𝑍 𝐵)
2518eleq2i 2690 . . . . . . . . . . . . . . . . 17 (𝑚𝑍𝑚 ∈ (ℤ𝑁))
2625biimpi 206 . . . . . . . . . . . . . . . 16 (𝑚𝑍𝑚 ∈ (ℤ𝑁))
27 eluzel2 11636 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (ℤ𝑁) → 𝑁 ∈ ℤ)
2826, 27syl 17 . . . . . . . . . . . . . . 15 (𝑚𝑍𝑁 ∈ ℤ)
29 eluzelz 11641 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (ℤ𝑁) → 𝑚 ∈ ℤ)
3026, 29syl 17 . . . . . . . . . . . . . . 15 (𝑚𝑍𝑚 ∈ ℤ)
3128, 30, 303jca 1240 . . . . . . . . . . . . . 14 (𝑚𝑍 → (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ))
32 eluzle 11644 . . . . . . . . . . . . . . 15 (𝑚 ∈ (ℤ𝑁) → 𝑁𝑚)
3326, 32syl 17 . . . . . . . . . . . . . 14 (𝑚𝑍𝑁𝑚)
3430zred 11426 . . . . . . . . . . . . . . 15 (𝑚𝑍𝑚 ∈ ℝ)
35 leid 10077 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℝ → 𝑚𝑚)
3634, 35syl 17 . . . . . . . . . . . . . 14 (𝑚𝑍𝑚𝑚)
3731, 33, 36jca32 557 . . . . . . . . . . . . 13 (𝑚𝑍 → ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑁𝑚𝑚𝑚)))
38 elfz2 12275 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑁...𝑚) ↔ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑁𝑚𝑚𝑚)))
3937, 38sylibr 224 . . . . . . . . . . . 12 (𝑚𝑍𝑚 ∈ (𝑁...𝑚))
40 nfcv 2761 . . . . . . . . . . . . . 14 𝑛𝑥
4140, 2nfel 2773 . . . . . . . . . . . . 13 𝑛 𝑥𝑚 / 𝑛𝐴
423eleq2d 2684 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝑥𝐴𝑥𝑚 / 𝑛𝐴))
4341, 42rspce 3290 . . . . . . . . . . . 12 ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥𝑚 / 𝑛𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
4439, 43sylan 488 . . . . . . . . . . 11 ((𝑚𝑍𝑥𝑚 / 𝑛𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
45 eliun 4490 . . . . . . . . . . 11 (𝑥 𝑛 ∈ (𝑁...𝑚)𝐴 ↔ ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
4644, 45sylibr 224 . . . . . . . . . 10 ((𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛 ∈ (𝑁...𝑚)𝐴)
47463adant2 1078 . . . . . . . . 9 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛 ∈ (𝑁...𝑚)𝐴)
4824, 47sseldd 3584 . . . . . . . 8 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛𝑍 𝐵)
4912, 14, 15, 48syl3anc 1323 . . . . . . 7 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛𝑍 𝐵)
50493exp 1261 . . . . . 6 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → (𝑚𝑍 → (𝑥𝑚 / 𝑛𝐴𝑥 𝑛𝑍 𝐵)))
5110, 11, 50rexlimd 3019 . . . . 5 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → (∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴𝑥 𝑛𝑍 𝐵))
5251adantr 481 . . . 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 2960 . 2 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑥 𝑛𝑍 𝐴𝑥 𝑛𝑍 𝐵)
55 dfss3 3573 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  csb 3514  wss 3555   ciun 4485   class class class wbr 4613  cfv 5847  (class class class)co 6604  cr 9879  cle 10019  cz 11321  cuz 11631  ...cfz 12268
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  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  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-iun 4487  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-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-neg 10213  df-z 11322  df-uz 11632  df-fz 12269
This theorem is referenced by:  iuneqfzuz  39012
  Copyright terms: Public domain W3C validator