Theorem iuneqfzuzlem 38491

Description: Lemma for iuneqfzuz 38492: 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   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.

Step	Hyp	Ref	Expression
1		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐴
2		nfcsb1v 3515	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
3		csbeq1a 3508	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
4	1, 2, 3	cbviun 4493	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴
5	4	eleq2i 2680	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ 𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴)
6		eliun 4460	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
7	5, 6	bitri 263	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
8	7	biimpi 205	. . . . 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 2925	. . . . . 6 ⊢ Ⅎ𝑚∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵
11		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑚 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵
12		simp2 1055	. . . . . . . 8 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑚 ∈ 𝑍)
13		rspa 2914	. . . . . . . . 9 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍) → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵)
14	13	3adant3 1074	. . . . . . . 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 1056	. . . . . . . 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 12253	. . . . . . . . . . . . 13 ⊢ (𝑁...𝑚) ⊆ (ℤ≥‘𝑁)
18		iuneqfzuzlem.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤ≥‘𝑁)
19	18	eqcomi 2619	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑁) = 𝑍
20	17, 19	sseqtri 3600	. . . . . . . . . . . 12 ⊢ (𝑁...𝑚) ⊆ 𝑍
21		iunss1 4468	. . . . . . . . . . . 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 3602	. . . . . . . . . 10 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
24	23	3ad2ant2 1076	. . . . . . . . 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 2680	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ 𝑍 ↔ 𝑚 ∈ (ℤ≥‘𝑁))
26	25	biimpi 205	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (ℤ≥‘𝑁))
27		eluzel2 11568	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ ℤ)
28	26, 27	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ 𝑍 → 𝑁 ∈ ℤ)
29		eluzelz 11573	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → 𝑚 ∈ ℤ)
30	26, 29	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ)
31	28, 30, 30	3jca 1235	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ 𝑍 → (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ))
32		eluzle 11576	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → 𝑁 ≤ 𝑚)
33	26, 32	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ 𝑍 → 𝑁 ≤ 𝑚)
34	30	zred 11358	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℝ)
35		leid 10012	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℝ → 𝑚 ≤ 𝑚)
36	34, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ≤ 𝑚)
37	31, 33, 36	jca32 556	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑍 → ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑁 ≤ 𝑚 ∧ 𝑚 ≤ 𝑚)))
38		elfz2 12204	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (𝑁...𝑚) ↔ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑁 ≤ 𝑚 ∧ 𝑚 ≤ 𝑚)))
39	37, 38	sylibr 223	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (𝑁...𝑚))
40		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛𝑥
41	40, 2	nfel 2763	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴
42	3	eleq2d 2673	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴))
43	41, 42	rspce 3277	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴)
44	39, 43	sylan 487	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴)
45		eliun 4460	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ↔ ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴)
46	44, 45	sylibr 223	. . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
47	46	3adant2 1073	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝑍 ∧ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
48	24, 47	sseldd 3569	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ∧ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
49	12, 14, 15, 48	syl3anc 1318	. . . . . . 7 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
50	49	3exp 1256	. . . . . 6 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (𝑚 ∈ 𝑍 → (𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)))
51	10, 11, 50	rexlimd 3008	. . . . 5 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵))
52	51	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))
53	9, 52	mpd 15	. . 3 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
54	53	ralrimiva 2949	. 2 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
55		dfss3 3558	. 2 ⊢ (∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵 ↔ ∀𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
56	54, 55	sylibr 223	1 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ⦋csb 3499   ⊆ wss 3540  ∪ ciun 4455   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814   ≤ cle 9954  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-neg 10148  df-z 11255  df-uz 11564  df-fz 12198

This theorem is referenced by:  iuneqfzuz  38492