Theorem fzuntd 44108

Description: Union of two adjacent finite sets of sequential integers that share a common endpoint. (Contributed by RP, 14-Dec-2024.)

Hypotheses

Ref	Expression
fzuntd.k	⊢ (𝜑 → 𝐾 ∈ ℤ)
fzuntd.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
fzuntd.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
fzuntd.km	⊢ (𝜑 → 𝐾 ≤ 𝑀)
fzuntd.mn	⊢ (𝜑 → 𝑀 ≤ 𝑁)

Assertion

Ref	Expression
fzuntd	⊢ (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))

Proof of Theorem fzuntd

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ∈ ℤ)
2	1	zred 12700	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ∈ ℝ)
3		fzuntd.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	zred 12700	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
5	4	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑀 ∈ ℝ)
6		fzuntd.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7	6	zred 12700	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8	7	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑁 ∈ ℝ)
9		simprr 784	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ≤ 𝑀)
10		fzuntd.mn	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
11	10	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑀 ≤ 𝑁)
12	2, 5, 8, 9, 11	letrd 11367	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ≤ 𝑁)
13	12	expr 461	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗 ≤ 𝑀 → 𝑗 ≤ 𝑁))
14	13	anim2d 623	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
15		fzuntd.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℤ)
16	15	zred 12700	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
17	16	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ∈ ℝ)
18	4	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑀 ∈ ℝ)
19		simprl 782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑗 ∈ ℤ)
20	19	zred 12700	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑗 ∈ ℝ)
21		fzuntd.km	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≤ 𝑀)
22	21	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ≤ 𝑀)
23		simprr 784	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑀 ≤ 𝑗)
24	17, 18, 20, 22, 23	letrd 11367	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ≤ 𝑗)
25	24	expr 461	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑀 ≤ 𝑗 → 𝐾 ≤ 𝑗))
26	25	anim1d 622	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
27	14, 26	jaod 872	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
28		orc 880	. . . . . . . . 9 ⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗))
29		orc 880	. . . . . . . . 9 ⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁))
30	28, 29	jca 520	. . . . . . . 8 ⊢ (𝐾 ≤ 𝑗 → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)))
31	30	ad2antrl 740	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)))
32		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ)
33	32	zred 12700	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℝ)
34	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑀 ∈ ℝ)
35	33, 34	letrid 11362	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗))
36	35	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗))
37		simprr 784	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → 𝑗 ≤ 𝑁)
38	37	olcd 887	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝑀 ∨ 𝑗 ≤ 𝑁))
39	36, 38	jca 520	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗) ∧ (𝑗 ≤ 𝑀 ∨ 𝑗 ≤ 𝑁)))
40		orddi 1025	. . . . . . 7 ⊢ (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)) ∧ ((𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗) ∧ (𝑗 ≤ 𝑀 ∨ 𝑗 ≤ 𝑁))))
41	31, 39, 40	sylanbrc 594	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
42	41	ex 417	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
43	27, 42	impbid 215	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
44	43	pm5.32da 589	. . 3 ⊢ (𝜑 → ((𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
45		elfz1 13540	. . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
46	15, 3, 45	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
47		3anass 1109	. . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
48	46, 47	bitrdi 290	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀))))
49		elfz1 13540	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
50	3, 6, 49	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
51		3anass 1109	. . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
52	50, 51	bitrdi 290	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
53	48, 52	orbi12d 931	. . . 4 ⊢ (𝜑 → ((𝑗 ∈ (𝐾...𝑀) ∨ 𝑗 ∈ (𝑀...𝑁)) ↔ ((𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)) ∨ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))))
54		elun 4115	. . . 4 ⊢ (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ (𝐾...𝑀) ∨ 𝑗 ∈ (𝑀...𝑁)))
55		andi 1023	. . . 4 ⊢ ((𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) ↔ ((𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)) ∨ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
56	53, 54, 55	3bitr4g 317	. . 3 ⊢ (𝜑 → (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))))
57		elfz1 13540	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
58	15, 6, 57	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
59		3anass 1109	. . . 4 ⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
60	58, 59	bitrdi 290	. . 3 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
61	44, 56, 60	3bitr4d 314	. 2 ⊢ (𝜑 → (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ 𝑗 ∈ (𝐾...𝑁)))
62	61	eqrdv 2767	1 ⊢ (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ∪ cun 3911   class class class wbr 5113  (class class class)co 7411  ℝcr 11099   ≤ cle 11244  ℤcz 12591  ...cfz 13535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-pre-lttri 11174  ax-pre-lttrn 11175

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-neg 11444  df-z 12592  df-fz 13536

This theorem is referenced by: (None)