Theorem iuneqfzuzlem 45915

Description: Lemma for iuneqfzuz 45916: 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 2926	. . . . . . . 8 ⊢ Ⅎ𝑚𝐴
2		nfcsb1v 3878	. . . . . . . 8 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
3		csbeq1a 3868	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
4	1, 2, 3	cbviun 4994	. . . . . . 7 ⊢ ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴
5	4	eleq2i 2856	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ 𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴)
6		eliun 4955	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
7	5, 6	bitri 277	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
8	7	bilani 508	. . . 4 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴) → ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
9		nfra1 3288	. . . . . 6 ⊢ Ⅎ𝑚∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵
10		nfv 1936	. . . . . 6 ⊢ Ⅎ𝑚 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵
11		simp2 1151	. . . . . . . 8 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑚 ∈ 𝑍)
12		rspa 3253	. . . . . . . . 9 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍) → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵)
13	12	3adant3 1146	. . . . . . . 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)
14		simp3 1152	. . . . . . . 8 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)
15		id 22	. . . . . . . . . . 11 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵)
16		fzssuz 13572	. . . . . . . . . . . . 13 ⊢ (𝑁...𝑚) ⊆ (ℤ≥‘𝑁)
17		iuneqfzuzlem.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤ≥‘𝑁)
18	17	eqcomi 2773	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑁) = 𝑍
19	16, 18	sseqtri 3986	. . . . . . . . . . . 12 ⊢ (𝑁...𝑚) ⊆ 𝑍
20		iunss1 4966	. . . . . . . . . . . 12 ⊢ ((𝑁...𝑚) ⊆ 𝑍 → ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
21	19, 20	mp1i 13	. . . . . . . . . . 11 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
22	15, 21	eqsstrd 3972	. . . . . . . . . 10 ⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
23	22	3ad2ant2 1148	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝑍 ∧ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)
24	17	eleq2i 2856	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ 𝑍 ↔ 𝑚 ∈ (ℤ≥‘𝑁))
25	24	biimpi 218	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (ℤ≥‘𝑁))
26		eluzel2 12846	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ ℤ)
27	25, 26	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑍 → 𝑁 ∈ ℤ)
28		eluzelz 12851	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → 𝑚 ∈ ℤ)
29	25, 28	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ)
30		eluzle 12854	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → 𝑁 ≤ 𝑚)
31	25, 30	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑍 → 𝑁 ≤ 𝑚)
32	29	zred 12679	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℝ)
33		leid 11281	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℝ → 𝑚 ≤ 𝑚)
34	32, 33	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ≤ 𝑚)
35	27, 29, 29, 31, 34	elfzd 13522	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (𝑁...𝑚))
36		nfcv 2926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛𝑥
37	36, 2	nfel 2940	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴
38	3	eleq2d 2850	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴))
39	37, 38	rspce 3572	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴)
40	35, 39	sylan 589	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴)
41		eliun 4955	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ↔ ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴)
42	40, 41	sylibr 236	. . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
43	42	3adant2 1145	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝑍 ∧ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴)
44	23, 43	sseldd 3939	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ∧ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
45	11, 13, 14, 44	syl3anc 1392	. . . . . . 7 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
46	45	3exp 1133	. . . . . 6 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (𝑚 ∈ 𝑍 → (𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)))
47	9, 10, 46	rexlimd 3271	. . . . 5 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵))
48	47	adantr 484	. . . 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))
49	8, 48	mpd 15	. . 3 ⊢ ((∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
50	49	ralrimiva 3156	. 2 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
51		dfss3 3927	. 2 ⊢ (∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵 ↔ ∀𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵)
52	50, 51	sylibr 236	1 ⊢ (∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪ 𝑛 ∈ 𝑍 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088  ⦋csb 3854   ⊆ wss 3906  ∪ ciun 4951   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398  ℝcr 11074   ≤ cle 11219  ℤcz 12570  ℤ_≥cuz 12841  ...cfz 13514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-cnex 11131  ax-resscn 11132  ax-pre-lttri 11149

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-neg 11419  df-z 12571  df-uz 12842  df-fz 13515

This theorem is referenced by:  iuneqfzuz  45916