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Theorem iuneqfzuzlem 44030
Description: Lemma for iuneqfzuz 44031: 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 2903 . . . . . . . . 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 2825 . . . . . . 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 482 . . . 4 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥 𝑛𝑍 𝐴) → ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
10 nfra1 3281 . . . . . 6 𝑚𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵
11 nfv 1917 . . . . . 6 𝑚 𝑥 𝑛𝑍 𝐵
12 simp2 1137 . . . . . . . 8 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑚𝑍)
13 rspa 3245 . . . . . . . . 9 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍) → 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵)
14133adant3 1132 . . . . . . . 8 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵)
15 simp3 1138 . . . . . . . 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 13538 . . . . . . . . . . . . 13 (𝑁...𝑚) ⊆ (ℤ𝑁)
18 iuneqfzuzlem.z . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑁)
1918eqcomi 2741 . . . . . . . . . . . . 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 1134 . . . . . . . . 9 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑛 ∈ (𝑁...𝑚)𝐴 𝑛𝑍 𝐵)
2518eleq2i 2825 . . . . . . . . . . . . . . 15 (𝑚𝑍𝑚 ∈ (ℤ𝑁))
2625biimpi 215 . . . . . . . . . . . . . 14 (𝑚𝑍𝑚 ∈ (ℤ𝑁))
27 eluzel2 12823 . . . . . . . . . . . . . 14 (𝑚 ∈ (ℤ𝑁) → 𝑁 ∈ ℤ)
2826, 27syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑁 ∈ ℤ)
29 eluzelz 12828 . . . . . . . . . . . . . 14 (𝑚 ∈ (ℤ𝑁) → 𝑚 ∈ ℤ)
3026, 29syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑚 ∈ ℤ)
31 eluzle 12831 . . . . . . . . . . . . . 14 (𝑚 ∈ (ℤ𝑁) → 𝑁𝑚)
3226, 31syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑁𝑚)
3330zred 12662 . . . . . . . . . . . . . 14 (𝑚𝑍𝑚 ∈ ℝ)
34 leid 11306 . . . . . . . . . . . . . 14 (𝑚 ∈ ℝ → 𝑚𝑚)
3533, 34syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑚𝑚)
3628, 30, 30, 32, 35elfzd 13488 . . . . . . . . . . . 12 (𝑚𝑍𝑚 ∈ (𝑁...𝑚))
37 nfcv 2903 . . . . . . . . . . . . . 14 𝑛𝑥
3837, 2nfel 2917 . . . . . . . . . . . . 13 𝑛 𝑥𝑚 / 𝑛𝐴
393eleq2d 2819 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝑥𝐴𝑥𝑚 / 𝑛𝐴))
4038, 39rspce 3601 . . . . . . . . . . . 12 ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥𝑚 / 𝑛𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
4136, 40sylan 580 . . . . . . . . . . 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 1131 . . . . . . . . 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 1371 . . . . . . 7 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛𝑍 𝐵)
47463exp 1119 . . . . . 6 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → (𝑚𝑍 → (𝑥𝑚 / 𝑛𝐴𝑥 𝑛𝑍 𝐵)))
4810, 11, 47rexlimd 3263 . . . . 5 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → (∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴𝑥 𝑛𝑍 𝐵))
4948adantr 481 . . . 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 3146 . 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070  csb 3892  wss 3947   ciun 4996   class class class wbr 5147  cfv 6540  (class class class)co 7405  cr 11105  cle 11245  cz 12554  cuz 12818  ...cfz 13480
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-pre-lttri 11180
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  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 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-neg 11443  df-z 12555  df-uz 12819  df-fz 13481
This theorem is referenced by:  iuneqfzuz  44031
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