iccpartiun - Mathbox for Alexander van der Vekens

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Theorem iccpartiun 46102

Description: A half-open interval of extended reals is the union of the parts of its partition. (Contributed by AV, 18-Jul-2020.)

**Hypotheses**
Ref	Expression
iccpartiun.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
iccpartiun.p	⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))

**Assertion**
Ref	Expression
iccpartiun	⊢ (𝜑 → ((𝑃‘0)[,)(𝑃‘𝑀)) = ∪ 𝑖 ∈ (0..^𝑀)((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))

Distinct variable groups: 𝑖,𝑀 𝑃,𝑖 𝜑,𝑖

Proof of Theorem iccpartiun Dummy variables 𝑗 𝑘 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpartiun.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))
2		iccpartiun.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		iccelpart 46101	. . . . . 6 ⊢ (𝑀 ∈ ℕ → ∀𝑝 ∈ (RePart‘𝑀)(𝑥 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
4		fveq1 6891	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
5		fveq1 6891	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑀) = (𝑃‘𝑀))
6	4, 5	oveq12d 7427	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → ((𝑝‘0)[,)(𝑝‘𝑀)) = ((𝑃‘0)[,)(𝑃‘𝑀)))
7	6	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑥 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) ↔ 𝑥 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))))
8		fveq1 6891	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑖) = (𝑃‘𝑖))
9		fveq1 6891	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
10	8, 9	oveq12d 7427	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))
11	10	eleq2d 2820	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑥 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
12	11	rexbidv 3179	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
13	7, 12	imbi12d 345	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑥 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑥 ∈ ((𝑃‘0)[,)(𝑃‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))))
14	13	rspcva 3611	. . . . . . 7 ⊢ ((𝑃 ∈ (RePart‘𝑀) ∧ ∀𝑝 ∈ (RePart‘𝑀)(𝑥 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) → (𝑥 ∈ ((𝑃‘0)[,)(𝑃‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
15	14	expcom 415	. . . . . 6 ⊢ (∀𝑝 ∈ (RePart‘𝑀)(𝑥 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) → (𝑃 ∈ (RePart‘𝑀) → (𝑥 ∈ ((𝑃‘0)[,)(𝑃‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))))
16	2, 3, 15	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ (RePart‘𝑀) → (𝑥 ∈ ((𝑃‘0)[,)(𝑃‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))))
17	1, 16	mpd 15	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((𝑃‘0)[,)(𝑃‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
18		nnnn0 12479	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
19		0elfz 13598	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ₀ → 0 ∈ (0...𝑀))
20	2, 18, 19	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑀))
21	2, 1, 20	iccpartxr 46087	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ^*)
22		nn0fz0 13599	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ₀ ↔ 𝑀 ∈ (0...𝑀))
23	22	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ (0...𝑀))
24	2, 18, 23	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
25	2, 1, 24	iccpartxr 46087	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ^*)
26	21, 25	jca 513	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘0) ∈ ℝ^* ∧ (𝑃‘𝑀) ∈ ℝ^*))
27	26	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃‘0) ∈ ℝ^* ∧ (𝑃‘𝑀) ∈ ℝ^*))
28		elfzofz 13648	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
29	2, 1	iccpartgel 46097	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑗))
30		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 → (𝑃‘𝑗) = (𝑃‘𝑖))
31	30	breq2d 5161	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → ((𝑃‘0) ≤ (𝑃‘𝑗) ↔ (𝑃‘0) ≤ (𝑃‘𝑖)))
32	31	rspcva 3611	. . . . . . . 8 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ∀𝑗 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑗)) → (𝑃‘0) ≤ (𝑃‘𝑖))
33	28, 29, 32	syl2anr 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘0) ≤ (𝑃‘𝑖))
34		fzofzp1 13729	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
35	2, 1	iccpartleu 46096	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀))
36		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑘 = (𝑖 + 1) → (𝑃‘𝑘) = (𝑃‘(𝑖 + 1)))
37	36	breq1d 5159	. . . . . . . . 9 ⊢ (𝑘 = (𝑖 + 1) → ((𝑃‘𝑘) ≤ (𝑃‘𝑀) ↔ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑀)))
38	37	rspcva 3611	. . . . . . . 8 ⊢ (((𝑖 + 1) ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀)) → (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑀))
39	34, 35, 38	syl2anr 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑀))
40		icossico 13394	. . . . . . 7 ⊢ ((((𝑃‘0) ∈ ℝ^* ∧ (𝑃‘𝑀) ∈ ℝ^*) ∧ ((𝑃‘0) ≤ (𝑃‘𝑖) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑀))) → ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ⊆ ((𝑃‘0)[,)(𝑃‘𝑀)))
41	27, 33, 39, 40	syl12anc 836	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ⊆ ((𝑃‘0)[,)(𝑃‘𝑀)))
42	41	sseld 3982	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) → 𝑥 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))))
43	42	rexlimdva 3156	. . . 4 ⊢ (𝜑 → (∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) → 𝑥 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))))
44	17, 43	impbid 211	. . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝑃‘0)[,)(𝑃‘𝑀)) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
45		eliun 5002	. . 3 ⊢ (𝑥 ∈ ∪ 𝑖 ∈ (0..^𝑀)((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))
46	44, 45	bitr4di 289	. 2 ⊢ (𝜑 → (𝑥 ∈ ((𝑃‘0)[,)(𝑃‘𝑀)) ↔ 𝑥 ∈ ∪ 𝑖 ∈ (0..^𝑀)((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
47	46	eqrdv 2731	1 ⊢ (𝜑 → ((𝑃‘0)[,)(𝑃‘𝑀)) = ∪ 𝑖 ∈ (0..^𝑀)((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 ∪ ciun 4998 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 0cc0 11110 1c1 11111 + caddc 11113 ℝ^*cxr 11247 ≤ cle 11249 ℕcn 12212 ℕ₀cn0 12472 [,)cico 13326 ...cfz 13484 ..^cfzo 13627 RePartciccp 46081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-ico 13330 df-fz 13485 df-fzo 13628 df-iccp 46082

This theorem is referenced by: (None)

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