Theorem iuneqfzuzlem 45330

Description: Lemma for iuneqfzuz 45331: 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 = (ℤ≥‘𝑁)

Assertion

Ref	Expression
iuneqfzuzlem	⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 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.

Step	Hyp	Ref	Expression
1		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐴
2		nfcsb1v 3886	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
3		csbeq1a 3876	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
4	1, 2, 3	cbviun 5000	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴
5	4	eleq2i 2820	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ 𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴)
6		eliun 4959	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
7	5, 6	bitri 275	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
8	7	biimpi 216	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 → ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
9	8	adantl 481	. . . 4 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴) → ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
10		nfra1 3261	. . . . . 6 ⊢ Ⅎ𝑚∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵
11		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑚 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵
12		simp2 1137	. . . . . . . 8 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑚 ∈ 𝑍)
13		rspa 3226	. . . . . . . . 9 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍) → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵)
14	13	3adant3 1132	. . . . . . . 8 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵)
15		simp3 1138	. . . . . . . 8 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
16		id 22	. . . . . . . . . . 11 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵)
17		fzssuz 13526	. . . . . . . . . . . . 13 ⊢ (𝑁...𝑚) ⊆ (ℤ≥‘𝑁)
18		iuneqfzuzlem.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤ≥‘𝑁)
19	18	eqcomi 2738	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑁) = 𝑍
20	17, 19	sseqtri 3995	. . . . . . . . . . . 12 ⊢ (𝑁...𝑚) ⊆ 𝑍
21		iunss1 4970	. . . . . . . . . . . 12 ⊢ ((𝑁...𝑚) ⊆ 𝑍 → ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
22	20, 21	mp1i 13	. . . . . . . . . . 11 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
23	16, 22	eqsstrd 3981	. . . . . . . . . 10 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
24	23	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝑍 ∧ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
25	18	eleq2i 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ 𝑍 ↔ 𝑚 ∈ (ℤ≥‘𝑁))
26	25	biimpi 216	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (ℤ≥‘𝑁))
27		eluzel2 12798	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ ℤ)
28	26, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑍 → 𝑁 ∈ ℤ)
29		eluzelz 12803	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → 𝑚 ∈ ℤ)
30	26, 29	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ)
31		eluzle 12806	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → 𝑁 ≤ 𝑚)
32	26, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑍 → 𝑁 ≤ 𝑚)
33	30	zred 12638	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℝ)
34		leid 11270	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℝ → 𝑚 ≤ 𝑚)
35	33, 34	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ≤ 𝑚)
36	28, 30, 30, 32, 35	elfzd 13476	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (𝑁...𝑚))
37		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛𝑥
38	37, 2	nfel 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴
39	3	eleq2d 2814	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴))
40	38, 39	rspce 3577	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴)
41	36, 40	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴)
42		eliun 4959	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ↔ ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴)
43	41, 42	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
44	43	3adant2 1131	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝑍 ∧ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
45	24, 44	sseldd 3947	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ∧ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
46	12, 14, 15, 45	syl3anc 1373	. . . . . . 7 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
47	46	3exp 1119	. . . . . 6 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (𝑚 ∈ 𝑍 → (𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)))
48	10, 11, 47	rexlimd 3244	. . . . 5 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵))
49	48	adantr 480	. . . 4 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴) → (∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵))
50	9, 49	mpd 15	. . 3 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
51	50	ralrimiva 3125	. 2 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
52		dfss3 3935	. 2 ⊢ (∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵 ↔ ∀𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
53	51, 52	sylibr 234	1 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  ⦋csb 3862   ⊆ wss 3914  ∪ ciun 4955   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387  ℝcr 11067   ≤ cle 11209  ℤcz 12529  ℤ_≥cuz 12793  ...cfz 13468

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-pre-lttri 11142

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-neg 11408  df-z 12530  df-uz 12794  df-fz 13469

This theorem is referenced by:  iuneqfzuz  45331