Theorem fzuntd 43418

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 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ∈ ℤ)
2	1	zred 12747	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ∈ ℝ)
3		fzuntd.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	zred 12747	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑀 ∈ ℝ)
6		fzuntd.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7	6	zred 12747	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑁 ∈ ℝ)
9		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ≤ 𝑀)
10		fzuntd.mn	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
11	10	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑀 ≤ 𝑁)
12	2, 5, 8, 9, 11	letrd 11447	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ≤ 𝑁)
13	12	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗 ≤ 𝑀 → 𝑗 ≤ 𝑁))
14	13	anim2d 611	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
15		fzuntd.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℤ)
16	15	zred 12747	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ∈ ℝ)
18	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑀 ∈ ℝ)
19		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑗 ∈ ℤ)
20	19	zred 12747	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑗 ∈ ℝ)
21		fzuntd.km	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≤ 𝑀)
22	21	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ≤ 𝑀)
23		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑀 ≤ 𝑗)
24	17, 18, 20, 22, 23	letrd 11447	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ≤ 𝑗)
25	24	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑀 ≤ 𝑗 → 𝐾 ≤ 𝑗))
26	25	anim1d 610	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
27	14, 26	jaod 858	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
28		orc 866	. . . . . . . . 9 ⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗))
29		orc 866	. . . . . . . . 9 ⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁))
30	28, 29	jca 511	. . . . . . . 8 ⊢ (𝐾 ≤ 𝑗 → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)))
31	30	ad2antrl 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)))
32		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ)
33	32	zred 12747	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℝ)
34	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑀 ∈ ℝ)
35	33, 34	letrid 11442	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗))
36	35	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗))
37		simprr 772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → 𝑗 ≤ 𝑁)
38	37	olcd 873	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝑀 ∨ 𝑗 ≤ 𝑁))
39	36, 38	jca 511	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗) ∧ (𝑗 ≤ 𝑀 ∨ 𝑗 ≤ 𝑁)))
40		orddi 1010	. . . . . . 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 582	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
42	41	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
43	27, 42	impbid 212	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
44	43	pm5.32da 578	. . 3 ⊢ (𝜑 → ((𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
45		elfz1 13572	. . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
46	15, 3, 45	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
47		3anass 1095	. . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
48	46, 47	bitrdi 287	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀))))
49		elfz1 13572	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
50	3, 6, 49	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
51		3anass 1095	. . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
52	50, 51	bitrdi 287	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
53	48, 52	orbi12d 917	. . . 4 ⊢ (𝜑 → ((𝑗 ∈ (𝐾...𝑀) ∨ 𝑗 ∈ (𝑀...𝑁)) ↔ ((𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)) ∨ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))))
54		elun 4176	. . . 4 ⊢ (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ (𝐾...𝑀) ∨ 𝑗 ∈ (𝑀...𝑁)))
55		andi 1008	. . . 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 314	. . 3 ⊢ (𝜑 → (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))))
57		elfz1 13572	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
58	15, 6, 57	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
59		3anass 1095	. . . 4 ⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
60	58, 59	bitrdi 287	. . 3 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
61	44, 56, 60	3bitr4d 311	. 2 ⊢ (𝜑 → (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ 𝑗 ∈ (𝐾...𝑁)))
62	61	eqrdv 2738	1 ⊢ (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ∪ cun 3974   class class class wbr 5166  (class class class)co 7448  ℝcr 11183   ≤ cle 11325  ℤcz 12639  ...cfz 13567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-pre-lttri 11258  ax-pre-lttrn 11259

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-neg 11523  df-z 12640  df-fz 13568

This theorem is referenced by: (None)