iccpartlt - Mathbox for Alexander van der Vekens

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Theorem iccpartlt 47737

Description: If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 46429 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 13619	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑀) ↔ 𝑀 ∈ ℕ)
4	1, 3	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
5		iccpartimp 47730	. . . . . . 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 1374	. . . . . 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 6835	. . . . . 6 ⊢ (𝑀 = 1 → (𝑃‘𝑀) = (𝑃‘1))
10		1e0p1 12653	. . . . . . 7 ⊢ 1 = (0 + 1)
11	10	fveq2i 6838	. . . . . 6 ⊢ (𝑃‘1) = (𝑃‘(0 + 1))
12	9, 11	eqtrdi 2788	. . . . 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 5127	. . 3 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀))
15	14	ex 412	. 2 ⊢ (𝑀 = 1 → (𝜑 → (𝑃‘0) < (𝑃‘𝑀)))
16	1, 2	iccpartiltu 47735	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
17	1, 2	iccpartigtl 47736	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
18		1nn 12160	. . . . . . . . . 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 2934	. . . . . . . . . . 11 ⊢ (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
22	1	nnge1d 12197	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑀)
23		1red 11137	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
24	1	nnred 12164	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
25	23, 24	ltlend 11282	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < 𝑀 ↔ (1 ≤ 𝑀 ∧ 𝑀 ≠ 1)))
26	25	biimprd 248	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 ≤ 𝑀 ∧ 𝑀 ≠ 1) → 1 < 𝑀))
27	22, 26	mpand 696	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ≠ 1 → 1 < 𝑀))
28	21, 27	biimtrrid 243	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑀 = 1 → 1 < 𝑀))
29	28	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 < 𝑀)
30		elfzo1 13632	. . . . . . . . 9 ⊢ (1 ∈ (1..^𝑀) ↔ (1 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 1 < 𝑀))
31	19, 20, 29, 30	syl3anbrc 1345	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ (1..^𝑀))
32		fveq2 6835	. . . . . . . . . 10 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
33	32	breq2d 5111	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((𝑃‘0) < (𝑃‘𝑖) ↔ (𝑃‘0) < (𝑃‘1)))
34	33	rspcv 3573	. . . . . . . 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 5109	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((𝑃‘𝑖) < (𝑃‘𝑀) ↔ (𝑃‘1) < (𝑃‘𝑀)))
37	36	rspcv 3573	. . . . . . . . . 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 12412	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
40		0elfz 13544	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → 0 ∈ (0...𝑀))
41	1, 39, 40	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑀))
42	1, 2, 41	iccpartxr 47732	. . . . . . . . . . . 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 12421	. . . . . . . . . . . . . 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 13538	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0...𝑀) ↔ (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
51	46, 48, 49, 50	syl3anbrc 1345	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ (0...𝑀))
52	20, 44, 51	iccpartxr 47732	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘1) ∈ ℝ^*)
53		nn0fz0 13545	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ₀ ↔ 𝑀 ∈ (0...𝑀))
54	39, 53	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
55	1, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
56	1, 2, 55	iccpartxr 47732	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ^*)
57	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘𝑀) ∈ ℝ^*)
58		xrlttr 13058	. . . . . . . . . . 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 1374	. . . . . . . . . 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 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 class class class wbr 5099 ‘cfv 6493 (class class class)co 7360 ↑_m cmap 8767 0cc0 11030 1c1 11031 + caddc 11033 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 ℕcn 12149 ℕ₀cn0 12405 ...cfz 13427 ..^cfzo 13574 RePartciccp 47726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 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 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-fzo 13575 df-iccp 47727

This theorem is referenced by: iccpartltu 47738 iccpartgtl 47739 iccpartgt 47740

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