iccpartlt - Mathbox for Alexander van der Vekens

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Theorem iccpartlt 44876

Description: If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 43659 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 13427	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑀) ↔ 𝑀 ∈ ℕ)
4	1, 3	sylibr 233	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
5		iccpartimp 44869	. . . . . . 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 1370	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘0) < (𝑃‘(0 + 1))))
7	6	simprd 496	. . . . 5 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘(0 + 1)))
8	7	adantl 482	. . . 4 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘0) < (𝑃‘(0 + 1)))
9		fveq2 6774	. . . . . 6 ⊢ (𝑀 = 1 → (𝑃‘𝑀) = (𝑃‘1))
10		1e0p1 12479	. . . . . . 7 ⊢ 1 = (0 + 1)
11	10	fveq2i 6777	. . . . . 6 ⊢ (𝑃‘1) = (𝑃‘(0 + 1))
12	9, 11	eqtrdi 2794	. . . . 5 ⊢ (𝑀 = 1 → (𝑃‘𝑀) = (𝑃‘(0 + 1)))
13	12	adantr 481	. . . 4 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘𝑀) = (𝑃‘(0 + 1)))
14	8, 13	breqtrrd 5102	. . 3 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀))
15	14	ex 413	. 2 ⊢ (𝑀 = 1 → (𝜑 → (𝑃‘0) < (𝑃‘𝑀)))
16	1, 2	iccpartiltu 44874	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
17	1, 2	iccpartigtl 44875	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
18		1nn 11984	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
19	18	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ ℕ)
20	1	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ)
21		df-ne 2944	. . . . . . . . . . 11 ⊢ (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
22	1	nnge1d 12021	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑀)
23		1red 10976	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
24	1	nnred 11988	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
25	23, 24	ltlend 11120	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < 𝑀 ↔ (1 ≤ 𝑀 ∧ 𝑀 ≠ 1)))
26	25	biimprd 247	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 ≤ 𝑀 ∧ 𝑀 ≠ 1) → 1 < 𝑀))
27	22, 26	mpand 692	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ≠ 1 → 1 < 𝑀))
28	21, 27	syl5bir 242	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑀 = 1 → 1 < 𝑀))
29	28	imp 407	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 < 𝑀)
30		elfzo1 13437	. . . . . . . . 9 ⊢ (1 ∈ (1..^𝑀) ↔ (1 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 1 < 𝑀))
31	19, 20, 29, 30	syl3anbrc 1342	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ (1..^𝑀))
32		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
33	32	breq2d 5086	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((𝑃‘0) < (𝑃‘𝑖) ↔ (𝑃‘0) < (𝑃‘1)))
34	33	rspcv 3557	. . . . . . . 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 5084	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((𝑃‘𝑖) < (𝑃‘𝑀) ↔ (𝑃‘1) < (𝑃‘𝑀)))
37	36	rspcv 3557	. . . . . . . . . 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 12240	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
40		0elfz 13353	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → 0 ∈ (0...𝑀))
41	1, 39, 40	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑀))
42	1, 2, 41	iccpartxr 44871	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ^*)
43	42	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘0) ∈ ℝ^*)
44	2	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑃 ∈ (RePart‘𝑀))
45		1nn0 12249	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ₀)
49	22	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ≤ 𝑀)
50		elfz2nn0 13347	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0...𝑀) ↔ (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
51	46, 48, 49, 50	syl3anbrc 1342	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ (0...𝑀))
52	20, 44, 51	iccpartxr 44871	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘1) ∈ ℝ^*)
53		nn0fz0 13354	. . . . . . . . . . . . . . 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 44871	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ^*)
57	56	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘𝑀) ∈ ℝ^*)
58		xrlttr 12874	. . . . . . . . . . 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 1370	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (((𝑃‘0) < (𝑃‘1) ∧ (𝑃‘1) < (𝑃‘𝑀)) → (𝑃‘0) < (𝑃‘𝑀)))
60	59	expcomd 417	. . . . . . . . 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 413	. . . . 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 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 0cc0 10871 1c1 10872 + caddc 10874 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 ℕcn 11973 ℕ₀cn0 12233 ...cfz 13239 ..^cfzo 13382 RePartciccp 44865

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-iccp 44866

This theorem is referenced by: iccpartltu 44877 iccpartgtl 44878 iccpartgt 44879

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