iccpartlt - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartlt		Structured version Visualization version GIF version

Theorem iccpartlt 48709

Description: If there is a partition, then the lower bound is strictly less than the upper bound. Corresponds to fourierdlem11 47331 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 13858	. . . . . . . 8 ⊢ (0 ∈ (0..^𝑀) ↔ 𝑀 ∈ ℕ)
4	1, 3	sylibr 237	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
5		iccpartimp 48702	. . . . . . 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 1399	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘0) < (𝑃‘(0 + 1))))
7	6	simprd 501	. . . . 5 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘(0 + 1)))
8	7	adantl 487	. . . 4 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘0) < (𝑃‘(0 + 1)))
9		fveq2 6905	. . . . . 6 ⊢ (𝑀 = 1 → (𝑃‘𝑀) = (𝑃‘1))
10		1e0p1 12885	. . . . . . 7 ⊢ 1 = (0 + 1)
11	10	fveq2i 6908	. . . . . 6 ⊢ (𝑃‘1) = (𝑃‘(0 + 1))
12	9, 11	eqtrdi 2814	. . . . 5 ⊢ (𝑀 = 1 → (𝑃‘𝑀) = (𝑃‘(0 + 1)))
13	12	adantr 486	. . . 4 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘𝑀) = (𝑃‘(0 + 1)))
14	8, 13	breqtrrd 5136	. . 3 ⊢ ((𝑀 = 1 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀))
15	14	ex 418	. 2 ⊢ (𝑀 = 1 → (𝜑 → (𝑃‘0) < (𝑃‘𝑀)))
16	1, 2	iccpartiltu 48707	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
17	1, 2	iccpartigtl 48708	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
18		1nn 12370	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
19	18	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ ℕ)
20	1	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ)
21		df-ne 2960	. . . . . . . . . . 11 ⊢ (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
22	1	nnge1d 12410	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑀)
23		1red 11333	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
24	1	nnred 12374	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
25	23, 24	ltlend 11479	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 < 𝑀 ↔ (1 ≤ 𝑀 ∧ 𝑀 ≠ 1)))
26	25	biimprd 251	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 ≤ 𝑀 ∧ 𝑀 ≠ 1) → 1 < 𝑀))
27	22, 26	mpand 708	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 ≠ 1 → 1 < 𝑀))
28	21, 27	biimtrrid 246	. . . . . . . . . 10 ⊢ (𝜑 → (¬ 𝑀 = 1 → 1 < 𝑀))
29	28	imp 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 < 𝑀)
30		elfzo1 13871	. . . . . . . . 9 ⊢ (1 ∈ (1..^𝑀) ↔ (1 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 1 < 𝑀))
31	19, 20, 29, 30	syl3anbrc 1362	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ (1..^𝑀))
32		fveq2 6905	. . . . . . . . . 10 ⊢ (𝑖 = 1 → (𝑃‘𝑖) = (𝑃‘1))
33	32	breq2d 5118	. . . . . . . . 9 ⊢ (𝑖 = 1 → ((𝑃‘0) < (𝑃‘𝑖) ↔ (𝑃‘0) < (𝑃‘1)))
34	33	rspcv 3576	. . . . . . . 8 ⊢ (1 ∈ (1..^𝑀) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) → (𝑃‘0) < (𝑃‘1)))
35	31, 34	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) → (𝑃‘0) < (𝑃‘1)))
36	32	breq1d 5116	. . . . . . . . . . 11 ⊢ (𝑖 = 1 → ((𝑃‘𝑖) < (𝑃‘𝑀) ↔ (𝑃‘1) < (𝑃‘𝑀)))
37	36	rspcv 3576	. . . . . . . . . 10 ⊢ (1 ∈ (1..^𝑀) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → (𝑃‘1) < (𝑃‘𝑀)))
38	31, 37	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → (𝑃‘1) < (𝑃‘𝑀)))
39		nnnn0 12637	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
40		0elfz 13782	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → 0 ∈ (0...𝑀))
41	1, 39, 40	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑀))
42	1, 2, 41	iccpartxr 48704	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ^*)
43	42	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘0) ∈ ℝ^*)
44	2	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑃 ∈ (RePart‘𝑀))
45		1nn0 12646	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
46	45	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ ℕ₀)
47	1, 39	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
48	47	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ₀)
49	22	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ≤ 𝑀)
50		elfz2nn0 13776	. . . . . . . . . . . . 13 ⊢ (1 ∈ (0...𝑀) ↔ (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
51	46, 48, 49, 50	syl3anbrc 1362	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 1 ∈ (0...𝑀))
52	20, 44, 51	iccpartxr 48704	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘1) ∈ ℝ^*)
53		nn0fz0 13783	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ₀ ↔ 𝑀 ∈ (0...𝑀))
54	39, 53	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
55	1, 54	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
56	1, 2, 55	iccpartxr 48704	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ^*)
57	56	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘𝑀) ∈ ℝ^*)
58		xrlttr 13293	. . . . . . . . . . 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 1399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (((𝑃‘0) < (𝑃‘1) ∧ (𝑃‘1) < (𝑃‘𝑀)) → (𝑃‘0) < (𝑃‘𝑀)))
60	59	expcomd 422	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → ((𝑃‘1) < (𝑃‘𝑀) → ((𝑃‘0) < (𝑃‘1) → (𝑃‘0) < (𝑃‘𝑀))))
61	38, 60	syld 48	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → ((𝑃‘0) < (𝑃‘1) → (𝑃‘0) < (𝑃‘𝑀))))
62	61	com23 87	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → ((𝑃‘0) < (𝑃‘1) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) → (𝑃‘0) < (𝑃‘𝑀))))
63	35, 62	syld 48	. . . . . 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 418	. . . . 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 96	. . . 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 50	. . 3 ⊢ (𝜑 → (¬ 𝑀 = 1 → (𝑃‘0) < (𝑃‘𝑀)))
67	66	com12 33	. 2 ⊢ (¬ 𝑀 = 1 → (𝜑 → (𝑃‘0) < (𝑃‘𝑀)))
68	15, 67	pm2.61i 184	1 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 class class class wbr 5106 ‘cfv 6555 (class class class)co 7440 ↑_m cmap 8872 0cc0 11224 1c1 11225 + caddc 11227 ℝ^*cxr 11366 < clt 11367 ≤ cle 11368 ℕcn 12359 ℕ₀cn0 12630 ...cfz 13663 ..^cfzo 13812 RePartciccp 48698

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-er 8742 df-map 8874 df-en 8999 df-dom 9000 df-sdom 9001 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-n0 12631 df-z 12718 df-uz 12990 df-fz 13664 df-fzo 13813 df-iccp 48699

This theorem is used by: iccpartltu 48710 iccpartgtl 48711 iccpartgt 48712

W3C validator