iccpartlt - Mathbox for Alexander van der Vekens

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Theorem iccpartlt 44764

Description: If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 43549 in GS's mathbox. (Contributed by AV, 12-Jul-2020.)

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

**Assertion**
Ref	Expression
iccpartlt	⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))

Proof of Theorem iccpartlt Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpartgtprec.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2		iccpartgtprec.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))
3		lbfzo0 13355	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑀) ↔ 𝑀 ∈ ℕ)
4	1, 3	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
5		iccpartimp 44757	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘0) < (𝑃‘(0 + 1))))
6	1, 2, 4, 5	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘0) < (𝑃‘(0 + 1))))
7	6	simprd 495	. . . . 5 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘(0 + 1)))
8	7	adantl 481	. . . 4 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘0) < (𝑃‘(0 + 1)))
9		fveq2 6756	. . . . . 6 ⊢ (𝑀 = 1 → (𝑃‘𝑀) = (𝑃‘1))
10		1e0p1 12408	. . . . . . 7 ⊢ 1 = (0 + 1)
11	10	fveq2i 6759	. . . . . 6 ⊢ (𝑃‘1) = (𝑃‘(0 + 1))
12	9, 11	eqtrdi 2795	. . . . 5 ⊢ (𝑀 = 1 → (𝑃‘𝑀) = (𝑃‘(0 + 1)))
13	12	adantr 480	. . . 4 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘𝑀) = (𝑃‘(0 + 1)))
14	8, 13	breqtrrd 5098	. . 3 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀))
15	14	ex 412	. 2 ⊢ (𝑀 = 1 → (𝜑 → (𝑃‘0) < (𝑃‘𝑀)))
16	1, 2	iccpartiltu 44762	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
17	1, 2	iccpartigtl 44763	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
18		1nn 11914	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
19	18	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ ℕ)
20	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ)
21		df-ne 2943	. . . . . . . . . . 11 ⊢ (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
22	1	nnge1d 11951	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑀)
23		1red 10907	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
24	1	nnred 11918	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
25	23, 24	ltlend 11050	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < 𝑀 ↔ (1 ≤ 𝑀 ∧ 𝑀 ≠ 1)))
26	25	biimprd 247	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 ≤ 𝑀 ∧ 𝑀 ≠ 1) → 1 < 𝑀))
27	22, 26	mpand 691	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ≠ 1 → 1 < 𝑀))
28	21, 27	syl5bir 242	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑀 = 1 → 1 < 𝑀))
29	28	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 < 𝑀)
30		elfzo1 13365	. . . . . . . . 9 ⊢ (1 ∈ (1..^𝑀) ↔ (1 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 1 < 𝑀))
31	19, 20, 29, 30	syl3anbrc 1341	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ (1..^𝑀))
32		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
33	32	breq2d 5082	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((𝑃‘0) < (𝑃‘𝑖) ↔ (𝑃‘0) < (𝑃‘1)))
34	33	rspcv 3547	. . . . . . . 8 ⊢ (1 ∈ (1..^𝑀) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) → (𝑃‘0) < (𝑃‘1)))
35	31, 34	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) → (𝑃‘0) < (𝑃‘1)))
36	32	breq1d 5080	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((𝑃‘𝑖) < (𝑃‘𝑀) ↔ (𝑃‘1) < (𝑃‘𝑀)))
37	36	rspcv 3547	. . . . . . . . . 10 ⊢ (1 ∈ (1..^𝑀) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → (𝑃‘1) < (𝑃‘𝑀)))
38	31, 37	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → (𝑃‘1) < (𝑃‘𝑀)))
39		nnnn0 12170	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
40		0elfz 13282	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → 0 ∈ (0...𝑀))
41	1, 39, 40	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑀))
42	1, 2, 41	iccpartxr 44759	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ^*)
43	42	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘0) ∈ ℝ^*)
44	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑃 ∈ (RePart‘𝑀))
45		1nn0 12179	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
46	45	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ ℕ₀)
47	1, 39	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
48	47	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ₀)
49	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ≤ 𝑀)
50		elfz2nn0 13276	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0...𝑀) ↔ (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
51	46, 48, 49, 50	syl3anbrc 1341	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ (0...𝑀))
52	20, 44, 51	iccpartxr 44759	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘1) ∈ ℝ^*)
53		nn0fz0 13283	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ₀ ↔ 𝑀 ∈ (0...𝑀))
54	39, 53	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
55	1, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
56	1, 2, 55	iccpartxr 44759	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ^*)
57	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘𝑀) ∈ ℝ^*)
58		xrlttr 12803	. . . . . . . . . . 11 ⊢ (((𝑃‘0) ∈ ℝ^* ∧ (𝑃‘1) ∈ ℝ^* ∧ (𝑃‘𝑀) ∈ ℝ^*) → (((𝑃‘0) < (𝑃‘1) ∧ (𝑃‘1) < (𝑃‘𝑀)) → (𝑃‘0) < (𝑃‘𝑀)))
59	43, 52, 57, 58	syl3anc 1369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (((𝑃‘0) < (𝑃‘1) ∧ (𝑃‘1) < (𝑃‘𝑀)) → (𝑃‘0) < (𝑃‘𝑀)))
60	59	expcomd 416	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → ((𝑃‘1) < (𝑃‘𝑀) → ((𝑃‘0) < (𝑃‘1) → (𝑃‘0) < (𝑃‘𝑀))))
61	38, 60	syld 47	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → ((𝑃‘0) < (𝑃‘1) → (𝑃‘0) < (𝑃‘𝑀))))
62	61	com23 86	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → ((𝑃‘0) < (𝑃‘1) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → (𝑃‘0) < (𝑃‘𝑀))))
63	35, 62	syld 47	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → (𝑃‘0) < (𝑃‘𝑀))))
64	63	ex 412	. . . . 5 ⊢ (𝜑 → (¬ 𝑀 = 1 → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → (𝑃‘0) < (𝑃‘𝑀)))))
65	64	com24 95	. . . 4 ⊢ (𝜑 → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) → (¬ 𝑀 = 1 → (𝑃‘0) < (𝑃‘𝑀)))))
66	16, 17, 65	mp2d 49	. . 3 ⊢ (𝜑 → (¬ 𝑀 = 1 → (𝑃‘0) < (𝑃‘𝑀)))
67	66	com12 32	. 2 ⊢ (¬ 𝑀 = 1 → (𝜑 → (𝑃‘0) < (𝑃‘𝑀)))
68	15, 67	pm2.61i 182	1 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 0cc0 10802 1c1 10803 + caddc 10805 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 ℕcn 11903 ℕ₀cn0 12163 ...cfz 13168 ..^cfzo 13311 RePartciccp 44753

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-iccp 44754

This theorem is referenced by: iccpartltu 44765 iccpartgtl 44766 iccpartgt 44767

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