iccpartlt - Mathbox for Alexander van der Vekens

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Theorem iccpartlt 47913

Description: If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 46575 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 13649	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑀) ↔ 𝑀 ∈ ℕ)
4	1, 3	sylibr 236	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
5		iccpartimp 47906	. . . . . . 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 1380	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘0) < (𝑃‘(0 + 1))))
7	6	simprd 497	. . . . 5 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘(0 + 1)))
8	7	adantl 483	. . . 4 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘0) < (𝑃‘(0 + 1)))
9		fveq2 6831	. . . . . 6 ⊢ (𝑀 = 1 → (𝑃‘𝑀) = (𝑃‘1))
10		1e0p1 12681	. . . . . . 7 ⊢ 1 = (0 + 1)
11	10	fveq2i 6834	. . . . . 6 ⊢ (𝑃‘1) = (𝑃‘(0 + 1))
12	9, 11	eqtrdi 2792	. . . . 5 ⊢ (𝑀 = 1 → (𝑃‘𝑀) = (𝑃‘(0 + 1)))
13	12	adantr 482	. . . 4 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘𝑀) = (𝑃‘(0 + 1)))
14	8, 13	breqtrrd 5103	. . 3 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀))
15	14	ex 414	. 2 ⊢ (𝑀 = 1 → (𝜑 → (𝑃‘0) < (𝑃‘𝑀)))
16	1, 2	iccpartiltu 47911	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
17	1, 2	iccpartigtl 47912	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
18		1nn 12180	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
19	18	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ ℕ)
20	1	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ)
21		df-ne 2937	. . . . . . . . . . 11 ⊢ (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
22	1	nnge1d 12220	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑀)
23		1red 11140	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
24	1	nnred 12184	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
25	23, 24	ltlend 11286	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < 𝑀 ↔ (1 ≤ 𝑀 ∧ 𝑀 ≠ 1)))
26	25	biimprd 250	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 ≤ 𝑀 ∧ 𝑀 ≠ 1) → 1 < 𝑀))
27	22, 26	mpand 702	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ≠ 1 → 1 < 𝑀))
28	21, 27	biimtrrid 245	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑀 = 1 → 1 < 𝑀))
29	28	imp 408	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 < 𝑀)
30		elfzo1 13662	. . . . . . . . 9 ⊢ (1 ∈ (1..^𝑀) ↔ (1 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 1 < 𝑀))
31	19, 20, 29, 30	syl3anbrc 1351	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ (1..^𝑀))
32		fveq2 6831	. . . . . . . . . 10 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
33	32	breq2d 5087	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((𝑃‘0) < (𝑃‘𝑖) ↔ (𝑃‘0) < (𝑃‘1)))
34	33	rspcv 3558	. . . . . . . 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 5085	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((𝑃‘𝑖) < (𝑃‘𝑀) ↔ (𝑃‘1) < (𝑃‘𝑀)))
37	36	rspcv 3558	. . . . . . . . . 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 12439	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
40		0elfz 13573	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → 0 ∈ (0...𝑀))
41	1, 39, 40	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑀))
42	1, 2, 41	iccpartxr 47908	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ^*)
43	42	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘0) ∈ ℝ^*)
44	2	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑃 ∈ (RePart‘𝑀))
45		1nn0 12448	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ₀)
49	22	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ≤ 𝑀)
50		elfz2nn0 13567	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0...𝑀) ↔ (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
51	46, 48, 49, 50	syl3anbrc 1351	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ (0...𝑀))
52	20, 44, 51	iccpartxr 47908	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘1) ∈ ℝ^*)
53		nn0fz0 13574	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ₀ ↔ 𝑀 ∈ (0...𝑀))
54	39, 53	sylib 220	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
55	1, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
56	1, 2, 55	iccpartxr 47908	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ^*)
57	56	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘𝑀) ∈ ℝ^*)
58		xrlttr 13086	. . . . . . . . . . 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 1380	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (((𝑃‘0) < (𝑃‘1) ∧ (𝑃‘1) < (𝑃‘𝑀)) → (𝑃‘0) < (𝑃‘𝑀)))
60	59	expcomd 418	. . . . . . . . 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 414	. . . . 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 183	1 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 class class class wbr 5075 ‘cfv 6489 (class class class)co 7360 ↑_m cmap 8767 0cc0 11033 1c1 11034 + caddc 11036 ℝ^*cxr 11173 < clt 11174 ≤ cle 11175 ℕcn 12169 ℕ₀cn0 12432 ...cfz 13456 ..^cfzo 13603 RePartciccp 47902

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-n0 12433 df-z 12520 df-uz 12784 df-fz 13457 df-fzo 13604 df-iccp 47903

This theorem is referenced by: iccpartltu 47914 iccpartgtl 47915 iccpartgt 47916

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