signslema - Mathbox for Thierry Arnoux

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Theorem signslema 35237

Description: Computational part of ~? signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)

**Hypotheses**
Ref	Expression
signslema.1	⊢ (𝜑 → 𝐸 ∈ ℕ₀)
signslema.2	⊢ (𝜑 → 𝐹 ∈ ℕ₀)
signslema.3	⊢ (𝜑 → 𝐺 ∈ ℕ₀)
signslema.4	⊢ (𝜑 → 𝐻 ∈ ℕ₀)
signslema.5	⊢ (𝜑 → (𝐸 < 𝐺 ∧ ¬ 2 ∥ (𝐺 − 𝐸)))
signslema.6	⊢ (𝜑 → ((𝐻 − 𝐺) − (𝐹 − 𝐸)) ∈ {0, 2})

**Assertion**
Ref	Expression
signslema	⊢ (𝜑 → (𝐹 < 𝐻 ∧ ¬ 2 ∥ (𝐻 − 𝐹)))

**Proof of Theorem signslema**
Step	Hyp	Ref	Expression
1		signslema.5	. . . . . 6 ⊢ (𝜑 → (𝐸 < 𝐺 ∧ ¬ 2 ∥ (𝐺 − 𝐸)))
2	1	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐸 < 𝐺)
3	2	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → 𝐸 < 𝐺)
4		signslema.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
5	4	nn0cnd 12672	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ ℂ)
6		signslema.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ℕ₀)
7	6	nn0cnd 12672	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ℂ)
8	5, 7	subcld 11672	. . . . . . . 8 ⊢ (𝜑 → (𝐻 − 𝐹) ∈ ℂ)
9		signslema.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ ℕ₀)
10	9	nn0cnd 12672	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ ℂ)
11		signslema.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
12	11	nn0cnd 12672	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℂ)
13	10, 12	subcld 11672	. . . . . . . 8 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℂ)
14	8, 13	subeq0ad 11680	. . . . . . 7 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ↔ (𝐻 − 𝐹) = (𝐺 − 𝐸)))
15	14	biimpa 482	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐻 − 𝐹) = (𝐺 − 𝐸))
16	15	breq2d 5114	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (0 < (𝐻 − 𝐹) ↔ 0 < (𝐺 − 𝐸)))
17	6	nn0red 12671	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ℝ)
18	4	nn0red 12671	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ ℝ)
19	17, 18	posdifd 11906	. . . . . 6 ⊢ (𝜑 → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
20	19	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
21	11	nn0red 12671	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
22	9	nn0red 12671	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ ℝ)
23	21, 22	posdifd 11906	. . . . . 6 ⊢ (𝜑 → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
24	23	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
25	16, 20, 24	3bitr4rd 315	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐸 < 𝐺 ↔ 𝐹 < 𝐻))
26	3, 25	mpbid 235	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → 𝐹 < 𝐻)
27		0red 11314	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 ∈ ℝ)
28	22, 21	resubcld 11747	. . . . . 6 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℝ)
29	28	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐺 − 𝐸) ∈ ℝ)
30	18, 17	resubcld 11747	. . . . . 6 ⊢ (𝜑 → (𝐻 − 𝐹) ∈ ℝ)
31	30	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐻 − 𝐹) ∈ ℝ)
32	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 𝐸 < 𝐺)
33	23	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
34	32, 33	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 < (𝐺 − 𝐸))
35		2pos 12450	. . . . . . 7 ⊢ 0 < 2
36		breq2 5106	. . . . . . 7 ⊢ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 → (0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ↔ 0 < 2))
37	35, 36	mpbiri 261	. . . . . 6 ⊢ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 → 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸)))
38	28, 30	posdifd 11906	. . . . . . 7 ⊢ (𝜑 → ((𝐺 − 𝐸) < (𝐻 − 𝐹) ↔ 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸))))
39	38	biimpar 483	. . . . . 6 ⊢ ((𝜑 ∧ 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸))) → (𝐺 − 𝐸) < (𝐻 − 𝐹))
40	37, 39	sylan2 605	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐺 − 𝐸) < (𝐻 − 𝐹))
41	27, 29, 31, 34, 40	lttrd 11474	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 < (𝐻 − 𝐹))
42	19	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
43	41, 42	mpbird 260	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 𝐹 < 𝐻)
44	5, 10, 7, 12	sub4d 11721	. . . . 5 ⊢ (𝜑 → ((𝐻 − 𝐺) − (𝐹 − 𝐸)) = ((𝐻 − 𝐹) − (𝐺 − 𝐸)))
45		signslema.6	. . . . 5 ⊢ (𝜑 → ((𝐻 − 𝐺) − (𝐹 − 𝐸)) ∈ {0, 2})
46	44, 45	eqeltrrd 2861	. . . 4 ⊢ (𝜑 → ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ {0, 2})
47		ovex 7453	. . . . 5 ⊢ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ V
48	47	elpr 4605	. . . 4 ⊢ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ {0, 2} ↔ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ∨ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2))
49	46, 48	sylib 221	. . 3 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ∨ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2))
50	26, 43, 49	mpjaodan 973	. 2 ⊢ (𝜑 → 𝐹 < 𝐻)
51	1	simprd 501	. . . . 5 ⊢ (𝜑 → ¬ 2 ∥ (𝐺 − 𝐸))
52	51	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → ¬ 2 ∥ (𝐺 − 𝐸))
53	15	breq2d 5114	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (2 ∥ (𝐻 − 𝐹) ↔ 2 ∥ (𝐺 − 𝐸)))
54	52, 53	mtbird 328	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → ¬ 2 ∥ (𝐻 − 𝐹))
55		2z 12731	. . . . . . 7 ⊢ 2 ∈ ℤ
56	9	nn0zd 12721	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℤ)
57	11	nn0zd 12721	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℤ)
58	56, 57	zsubcld 12811	. . . . . . 7 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℤ)
59		dvdsaddr 16478	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ (𝐺 − 𝐸) ∈ ℤ) → (2 ∥ (𝐺 − 𝐸) ↔ 2 ∥ ((𝐺 − 𝐸) + 2)))
60	55, 58, 59	sylancr 599	. . . . . 6 ⊢ (𝜑 → (2 ∥ (𝐺 − 𝐸) ↔ 2 ∥ ((𝐺 − 𝐸) + 2)))
61	51, 60	mtbid 327	. . . . 5 ⊢ (𝜑 → ¬ 2 ∥ ((𝐺 − 𝐸) + 2))
62	61	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ¬ 2 ∥ ((𝐺 − 𝐸) + 2))
63		2cnd 12424	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
64	8, 13, 63	subaddd 11690	. . . . . 6 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 ↔ ((𝐺 − 𝐸) + 2) = (𝐻 − 𝐹)))
65	64	biimpa 482	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ((𝐺 − 𝐸) + 2) = (𝐻 − 𝐹))
66	65	breq2d 5114	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (2 ∥ ((𝐺 − 𝐸) + 2) ↔ 2 ∥ (𝐻 − 𝐹)))
67	62, 66	mtbid 327	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ¬ 2 ∥ (𝐻 − 𝐹))
68	54, 67, 49	mpjaodan 973	. 2 ⊢ (𝜑 → ¬ 2 ∥ (𝐻 − 𝐹))
69	50, 68	jca 521	1 ⊢ (𝜑 → (𝐹 < 𝐻 ∧ ¬ 2 ∥ (𝐻 − 𝐹)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 {cpr 4586 class class class wbr 5102 (class class class)co 7420 ℝcr 11202 0cc0 11203 + caddc 11206 < clt 11346 − cmin 11544 2c2 12400 ℕ₀cn0 12609 ℤcz 12696 ∥ cdvds 16427

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-er 8720 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-nn 12339 df-2 12408 df-n0 12610 df-z 12697 df-dvds 16428

This theorem is used by: (None)

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