Theorem fzuntd 43707

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 12596	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ∈ ℝ)
3		fzuntd.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4	3	zred 12596	. . . . . . . . . 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 12596	. . . . . . . . . 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 11290	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑗 ≤ 𝑀)) → 𝑗 ≤ 𝑁)
13	12	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗 ≤ 𝑀 → 𝑗 ≤ 𝑁))
14	13	anim2d 612	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
15		fzuntd.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℤ)
16	15	zred 12596	. . . . . . . . . 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 12596	. . . . . . . . 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 11290	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) → 𝐾 ≤ 𝑗)
25	24	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑀 ≤ 𝑗 → 𝐾 ≤ 𝑗))
26	25	anim1d 611	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
27	14, 26	jaod 859	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)) → (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
28		orc 867	. . . . . . . . 9 ⊢ (𝐾 ≤ 𝑗 → (𝐾 ≤ 𝑗 ∨ 𝑀 ≤ 𝑗))
29		orc 867	. . . . . . . . 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 728	. . . . . . 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 12596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℝ)
34	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑀 ∈ ℝ)
35	33, 34	letrid 11285	. . . . . . . . 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 874	. . . . . . . 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 1011	. . . . . . 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 583	. . . . . 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 579	. . 3 ⊢ (𝜑 → ((𝑗 ∈ ℤ ∧ ((𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀) ∨ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
45		elfz1 13428	. . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
46	15, 3, 45	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑀) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)))
47		3anass 1094	. . . . . 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 13428	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
50	3, 6, 49	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
51		3anass 1094	. . . . . 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 918	. . . 4 ⊢ (𝜑 → ((𝑗 ∈ (𝐾...𝑀) ∨ 𝑗 ∈ (𝑀...𝑁)) ↔ ((𝑗 ∈ ℤ ∧ (𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑀)) ∨ (𝑗 ∈ ℤ ∧ (𝑀 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))))
54		elun 4105	. . . 4 ⊢ (𝑗 ∈ ((𝐾...𝑀) ∪ (𝑀...𝑁)) ↔ (𝑗 ∈ (𝐾...𝑀) ∨ 𝑗 ∈ (𝑀...𝑁)))
55		andi 1009	. . . 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 13428	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
58	15, 6, 57	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑗 ∈ (𝐾...𝑁) ↔ (𝑗 ∈ ℤ ∧ 𝐾 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
59		3anass 1094	. . . 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 2734	1 ⊢ (𝜑 → ((𝐾...𝑀) ∪ (𝑀...𝑁)) = (𝐾...𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ∪ cun 3899   class class class wbr 5098  (class class class)co 7358  ℝcr 11025   ≤ cle 11167  ℤcz 12488  ...cfz 13423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-pre-lttri 11100  ax-pre-lttrn 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-neg 11367  df-z 12489  df-fz 13424

This theorem is referenced by: (None)