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Theorem iuneqfzuzlem 45283
Description: Lemma for iuneqfzuz 45284: 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 2902 . . . . . . . . 9 𝑚𝐴
2 nfcsb1v 3932 . . . . . . . . 9 𝑛𝑚 / 𝑛𝐴
3 csbeq1a 3921 . . . . . . . . 9 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
41, 2, 3cbviun 5040 . . . . . . . 8 𝑛𝑍 𝐴 = 𝑚𝑍 𝑚 / 𝑛𝐴
54eleq2i 2830 . . . . . . 7 (𝑥 𝑛𝑍 𝐴𝑥 𝑚𝑍 𝑚 / 𝑛𝐴)
6 eliun 4999 . . . . . . 7 (𝑥 𝑚𝑍 𝑚 / 𝑛𝐴 ↔ ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
75, 6bitri 275 . . . . . 6 (𝑥 𝑛𝑍 𝐴 ↔ ∃𝑚𝑍 𝑥𝑚 / 𝑛𝐴)
87biimpi 216 . . . . 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 3281 . . . . . 6 𝑚𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵
11 nfv 1911 . . . . . 6 𝑚 𝑥 𝑛𝑍 𝐵
12 simp2 1136 . . . . . . . 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 1131 . . . . . . . 8 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵)
15 simp3 1137 . . . . . . . 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 13601 . . . . . . . . . . . . 13 (𝑁...𝑚) ⊆ (ℤ𝑁)
18 iuneqfzuzlem.z . . . . . . . . . . . . . 14 𝑍 = (ℤ𝑁)
1918eqcomi 2743 . . . . . . . . . . . . 13 (ℤ𝑁) = 𝑍
2017, 19sseqtri 4031 . . . . . . . . . . . 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 4033 . . . . . . . . . 10 ( 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 𝑛 ∈ (𝑁...𝑚)𝐴 𝑛𝑍 𝐵)
24233ad2ant2 1133 . . . . . . . . 9 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑛 ∈ (𝑁...𝑚)𝐴 𝑛𝑍 𝐵)
2518eleq2i 2830 . . . . . . . . . . . . . . 15 (𝑚𝑍𝑚 ∈ (ℤ𝑁))
2625biimpi 216 . . . . . . . . . . . . . 14 (𝑚𝑍𝑚 ∈ (ℤ𝑁))
27 eluzel2 12880 . . . . . . . . . . . . . 14 (𝑚 ∈ (ℤ𝑁) → 𝑁 ∈ ℤ)
2826, 27syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑁 ∈ ℤ)
29 eluzelz 12885 . . . . . . . . . . . . . 14 (𝑚 ∈ (ℤ𝑁) → 𝑚 ∈ ℤ)
3026, 29syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑚 ∈ ℤ)
31 eluzle 12888 . . . . . . . . . . . . . 14 (𝑚 ∈ (ℤ𝑁) → 𝑁𝑚)
3226, 31syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑁𝑚)
3330zred 12719 . . . . . . . . . . . . . 14 (𝑚𝑍𝑚 ∈ ℝ)
34 leid 11354 . . . . . . . . . . . . . 14 (𝑚 ∈ ℝ → 𝑚𝑚)
3533, 34syl 17 . . . . . . . . . . . . 13 (𝑚𝑍𝑚𝑚)
3628, 30, 30, 32, 35elfzd 13551 . . . . . . . . . . . 12 (𝑚𝑍𝑚 ∈ (𝑁...𝑚))
37 nfcv 2902 . . . . . . . . . . . . . 14 𝑛𝑥
3837, 2nfel 2917 . . . . . . . . . . . . 13 𝑛 𝑥𝑚 / 𝑛𝐴
393eleq2d 2824 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝑥𝐴𝑥𝑚 / 𝑛𝐴))
4038, 39rspce 3610 . . . . . . . . . . . 12 ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥𝑚 / 𝑛𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
4136, 40sylan 580 . . . . . . . . . . 11 ((𝑚𝑍𝑥𝑚 / 𝑛𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
42 eliun 4999 . . . . . . . . . . 11 (𝑥 𝑛 ∈ (𝑁...𝑚)𝐴 ↔ ∃𝑛 ∈ (𝑁...𝑚)𝑥𝐴)
4341, 42sylibr 234 . . . . . . . . . 10 ((𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛 ∈ (𝑁...𝑚)𝐴)
44433adant2 1130 . . . . . . . . 9 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛 ∈ (𝑁...𝑚)𝐴)
4524, 44sseldd 3995 . . . . . . . 8 ((𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛𝑍 𝐵)
4612, 14, 15, 45syl3anc 1370 . . . . . . 7 ((∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵𝑚𝑍𝑥𝑚 / 𝑛𝐴) → 𝑥 𝑛𝑍 𝐵)
47463exp 1118 . . . . . 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 480 . . . 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 3143 . 2 (∀𝑚𝑍 𝑛 ∈ (𝑁...𝑚)𝐴 = 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑥 𝑛𝑍 𝐴𝑥 𝑛𝑍 𝐵)
52 dfss3 3983 . 2 ( 𝑛𝑍 𝐴 𝑛𝑍 𝐵 ↔ ∀𝑥 𝑛𝑍 𝐴𝑥 𝑛𝑍 𝐵)
5351, 52sylibr 234 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 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  csb 3907  wss 3962   ciun 4995   class class class wbr 5147  cfv 6562  (class class class)co 7430  cr 11151  cle 11293  cz 12610  cuz 12875  ...cfz 13543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-pre-lttri 11226
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-neg 11492  df-z 12611  df-uz 12876  df-fz 13544
This theorem is referenced by:  iuneqfzuz  45284
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