Theorem fzuntd 42889

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 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ∈ ℤ)
2	1	zred 12702	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ∈ ℝ)
3		fzuntd.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	zred 12702	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
5	4	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑀 ∈ ℝ)
6		fzuntd.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7	6	zred 12702	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
8	7	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑁 ∈ ℝ)
9		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ≤ 𝑀)
10		fzuntd.mn	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
11	10	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑀 ≤ 𝑁)
12	2, 5, 8, 9, 11	letrd 11407	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ≤ 𝑁)
13	12	expr 455	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗 ≤ 𝑀 → 𝑗 ≤ 𝑁))
14	13	anim2d 610	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
15		fzuntd.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℤ)
16	15	zred 12702	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
17	16	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ∈ ℝ)
18	4	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑀 ∈ ℝ)
19		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑗 ∈ ℤ)
20	19	zred 12702	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑗 ∈ ℝ)
21		fzuntd.km	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ≤ 𝑀)
22	21	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ≤ 𝑀)
23		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝑀 ≤ 𝑗)
24	17, 18, 20, 22, 23	letrd 11407	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ≤ 𝑗)
25	24	expr 455	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑀 ≤ 𝑗 → 𝐾 ≤ 𝑗))
26	25	anim1d 609	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
27	14, 26	jaod 857	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
28		orc 865	. . . . . . . . 9 ⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗))
29		orc 865	. . . . . . . . 9 ⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁))
30	28, 29	jca 510	. . . . . . . 8 ⊢ (𝐾 ≤ 𝑗 → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)))
31	30	ad2antrl 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗) ∧ (𝐾 ≤ 𝑗 ∨ 𝑗 ≤ 𝑁)))
32		simpr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ)
33	32	zred 12702	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℝ)
34	4	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑀 ∈ ℝ)
35	33, 34	letrid 11402	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗))
36	35	adantr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗))
37		simprr 771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → 𝑗 ≤ 𝑁)
38	37	olcd 872	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝑗 ≤ 𝑀 ∨ 𝑗 ≤ 𝑁))
39	36, 38	jca 510	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝑗 ≤ 𝑀 ∨ 𝑀 ≤ 𝑗) ∧ (𝑗 ≤ 𝑀 ∨ 𝑗 ≤ 𝑁)))
40		orddi 1007	. . . . . . 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 581	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℤ) ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
42	41	ex 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
43	27, 42	impbid 211	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) ↔ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
44	43	pm5.32da 577	. . 3 ⊢ (𝜑 → ((𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
45		elfz1 13527	. . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
46	15, 3, 45	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
47		3anass 1092	. . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
48	46, 47	bitrdi 286	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀))))
49		elfz1 13527	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
50	3, 6, 49	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
51		3anass 1092	. . . . . 6 ⊢ ((𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
52	50, 51	bitrdi 286	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
53	48, 52	orbi12d 916	. . . 4 ⊢ (𝜑 → ((𝑗 ∈ (𝐾...𝑀) ∨ 𝑗 ∈ (𝑀...𝑁)) ↔ ((𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)) ∨ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))))
54		elun 4147	. . . 4 ⊢ (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ (𝐾...𝑀) ∨ 𝑗 ∈ (𝑀...𝑁)))
55		andi 1005	. . . 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 313	. . 3 ⊢ (𝜑 → (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))))
57		elfz1 13527	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
58	15, 6, 57	syl2anc 582	. . . 4 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
59		3anass 1092	. . . 4 ⊢ ((𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
60	58, 59	bitrdi 286	. . 3 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
61	44, 56, 60	3bitr4d 310	. 2 ⊢ (𝜑 → (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ 𝑗 ∈ (𝐾...𝑁)))
62	61	eqrdv 2725	1 ⊢ (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∪ cun 3945   class class class wbr 5150  (class class class)co 7424  ℝcr 11143   ≤ cle 11285  ℤcz 12594  ...cfz 13522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-pre-lttri 11218  ax-pre-lttrn 11219

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-er 8729  df-en 8969  df-dom 8970  df-sdom 8971  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-neg 11483  df-z 12595  df-fz 13523

This theorem is referenced by: (None)