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Theorem iuneqfzuzlem 44342
Description: Lemma for iuneqfzuz 44343: 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 2901 . . . . . . . . 9 𝑚𝐴
2 nfcsb1v 3917 . . . . . . . . 9 𝑛𝑚 / 𝑛𝐴
3 csbeq1a 3906 . . . . . . . . 9 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
41, 2, 3cbviun 5038 . . . . . . . 8 𝑛𝑍 𝐴 = 𝑚𝑍 𝑚 / 𝑛𝐴
54eleq2i 2823 . . . . . . 7 (𝑥 𝑛𝑍 𝐴𝑥 𝑚𝑍 𝑚 / 𝑛𝐴)
6 eliun 5000 . . . . . . 7 (𝑥 𝑚𝑍 𝑚 / 𝑛𝐴 ↔ ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
75, 6bitri 274 . . . . . 6 (𝑥 𝑛𝑍 𝐴 ↔ ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
87biimpi 215 . . . . 5 (𝑥 𝑛𝑍 𝐴 → ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
98adantl 480 . . . 4 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥 𝑛𝑍 𝐴) → ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
10 nfra1 3279 . . . . . 6 𝑚𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵
11 nfv 1915 . . . . . 6 𝑚 𝑥 𝑛𝑍 𝐵
12 simp2 1135 . . . . . . . 8 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑚𝑍)
13 rspa 3243 . . . . . . . . 9 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍) → 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵)
14133adant3 1130 . . . . . . . 8 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵)
15 simp3 1136 . . . . . . . 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 13546 . . . . . . . . . . . . 13 (𝑁...𝑚) ⊆ (ℤ𝑁)
18 iuneqfzuzlem.z . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑁)
1918eqcomi 2739 . . . . . . . . . . . . 13 (ℤ𝑁) = 𝑍
2017, 19sseqtri 4017 . . . . . . . . . . . 12 (𝑁...𝑚) ⊆ 𝑍
21 iunss1 5010 . . . . . . . . . . . 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 4019 . . . . . . . . . 10 ( 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛 ∈ (𝑁...𝑚)𝐴 𝑛𝑍 𝐵)
24233ad2ant2 1132 . . . . . . . . 9 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑛 ∈ (𝑁...𝑚)𝐴 𝑛𝑍 𝐵)
2518eleq2i 2823 . . . . . . . . . . . . . . 15 (𝑚𝑍𝑚 ∈ (ℤ𝑁))
2625biimpi 215 . . . . . . . . . . . . . 14 (𝑚𝑍𝑚 ∈ (ℤ𝑁))
27 eluzel2 12831 . . . . . . . . . . . . . 14 (𝑚 ∈ (ℤ𝑁) → 𝑁 ∈ ℤ)
2826, 27syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑁 ∈ ℤ)
29 eluzelz 12836 . . . . . . . . . . . . . 14 (𝑚 ∈ (ℤ𝑁) → 𝑚 ∈ ℤ)
3026, 29syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑚 ∈ ℤ)
31 eluzle 12839 . . . . . . . . . . . . . 14 (𝑚 ∈ (ℤ𝑁) → 𝑁𝑚)
3226, 31syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑁𝑚)
3330zred 12670 . . . . . . . . . . . . . 14 (𝑚𝑍𝑚 ∈ ℝ)
34 leid 11314 . . . . . . . . . . . . . 14 (𝑚 ∈ ℝ → 𝑚𝑚)
3533, 34syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑚𝑚)
3628, 30, 30, 32, 35elfzd 13496 . . . . . . . . . . . 12 (𝑚𝑍𝑚 ∈ (𝑁...𝑚))
37 nfcv 2901 . . . . . . . . . . . . . 14 𝑛𝑥
3837, 2nfel 2915 . . . . . . . . . . . . 13 𝑛 𝑥𝑚 / 𝑛𝐴
393eleq2d 2817 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝑥𝐴𝑥𝑚 / 𝑛𝐴))
4038, 39rspce 3600 . . . . . . . . . . . 12 ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥𝑚 / 𝑛𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
4136, 40sylan 578 . . . . . . . . . . 11 ((𝑚𝑍𝑥𝑚 / 𝑛𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
42 eliun 5000 . . . . . . . . . . 11 (𝑥 𝑛 ∈ (𝑁...𝑚)𝐴 ↔ ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
4341, 42sylibr 233 . . . . . . . . . 10 ((𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛 ∈ (𝑁...𝑚)𝐴)
44433adant2 1129 . . . . . . . . 9 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛 ∈ (𝑁...𝑚)𝐴)
4524, 44sseldd 3982 . . . . . . . 8 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛𝑍 𝐵)
4612, 14, 15, 45syl3anc 1369 . . . . . . 7 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛𝑍 𝐵)
47463exp 1117 . . . . . 6 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → (𝑚𝑍 → (𝑥𝑚 / 𝑛𝐴𝑥 𝑛𝑍 𝐵)))
4810, 11, 47rexlimd 3261 . . . . 5 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → (∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴𝑥 𝑛𝑍 𝐵))
4948adantr 479 . . . 4 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥 𝑛𝑍 𝐴) → (∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴𝑥 𝑛𝑍 𝐵))
509, 49mpd 15 . . 3 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥 𝑛𝑍 𝐴) → 𝑥 𝑛𝑍 𝐵)
5150ralrimiva 3144 . 2 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑥 𝑛𝑍 𝐴𝑥 𝑛𝑍 𝐵)
52 dfss3 3969 . 2 ( 𝑛𝑍 𝐴 𝑛𝑍 𝐵 ↔ ∀𝑥 𝑛𝑍 𝐴𝑥 𝑛𝑍 𝐵)
5351, 52sylibr 233 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 394  w3a 1085   = wceq 1539  wcel 2104  wral 3059  wrex 3068  csb 3892  wss 3947   ciun 4996   class class class wbr 5147  cfv 6542  (class class class)co 7411  cr 11111  cle 11253  cz 12562  cuz 12826  ...cfz 13488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-pre-lttri 11186
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-neg 11451  df-z 12563  df-uz 12827  df-fz 13489
This theorem is referenced by:  iuneqfzuz  44343
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