signslema - Mathbox for Thierry Arnoux

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Theorem signslema 33573

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 496	. . . . 5 ⊢ (𝜑 → 𝐸 < 𝐺)
3	2	adantr 482	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → 𝐸 < 𝐺)
4		signslema.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
5	4	nn0cnd 12534	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ ℂ)
6		signslema.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ℕ₀)
7	6	nn0cnd 12534	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ℂ)
8	5, 7	subcld 11571	. . . . . . . 8 ⊢ (𝜑 → (𝐻 − 𝐹) ∈ ℂ)
9		signslema.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ ℕ₀)
10	9	nn0cnd 12534	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ ℂ)
11		signslema.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
12	11	nn0cnd 12534	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℂ)
13	10, 12	subcld 11571	. . . . . . . 8 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℂ)
14	8, 13	subeq0ad 11581	. . . . . . 7 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ↔ (𝐻 − 𝐹) = (𝐺 − 𝐸)))
15	14	biimpa 478	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐻 − 𝐹) = (𝐺 − 𝐸))
16	15	breq2d 5161	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (0 < (𝐻 − 𝐹) ↔ 0 < (𝐺 − 𝐸)))
17	6	nn0red 12533	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ℝ)
18	4	nn0red 12533	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ ℝ)
19	17, 18	posdifd 11801	. . . . . 6 ⊢ (𝜑 → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
20	19	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
21	11	nn0red 12533	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
22	9	nn0red 12533	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ ℝ)
23	21, 22	posdifd 11801	. . . . . 6 ⊢ (𝜑 → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
24	23	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
25	16, 20, 24	3bitr4rd 312	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐸 < 𝐺 ↔ 𝐹 < 𝐻))
26	3, 25	mpbid 231	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → 𝐹 < 𝐻)
27		0red 11217	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 ∈ ℝ)
28	22, 21	resubcld 11642	. . . . . 6 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℝ)
29	28	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐺 − 𝐸) ∈ ℝ)
30	18, 17	resubcld 11642	. . . . . 6 ⊢ (𝜑 → (𝐻 − 𝐹) ∈ ℝ)
31	30	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐻 − 𝐹) ∈ ℝ)
32	2	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 𝐸 < 𝐺)
33	23	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
34	32, 33	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 < (𝐺 − 𝐸))
35		2pos 12315	. . . . . . 7 ⊢ 0 < 2
36		breq2 5153	. . . . . . 7 ⊢ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 → (0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ↔ 0 < 2))
37	35, 36	mpbiri 258	. . . . . 6 ⊢ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 → 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸)))
38	28, 30	posdifd 11801	. . . . . . 7 ⊢ (𝜑 → ((𝐺 − 𝐸) < (𝐻 − 𝐹) ↔ 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸))))
39	38	biimpar 479	. . . . . 6 ⊢ ((𝜑 ∧ 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸))) → (𝐺 − 𝐸) < (𝐻 − 𝐹))
40	37, 39	sylan2 594	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐺 − 𝐸) < (𝐻 − 𝐹))
41	27, 29, 31, 34, 40	lttrd 11375	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 < (𝐻 − 𝐹))
42	19	adantr 482	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
43	41, 42	mpbird 257	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 𝐹 < 𝐻)
44	5, 10, 7, 12	sub4d 11620	. . . . 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 2835	. . . 4 ⊢ (𝜑 → ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ {0, 2})
47		ovex 7442	. . . . 5 ⊢ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ V
48	47	elpr 4652	. . . 4 ⊢ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ {0, 2} ↔ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ∨ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2))
49	46, 48	sylib 217	. . 3 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ∨ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2))
50	26, 43, 49	mpjaodan 958	. 2 ⊢ (𝜑 → 𝐹 < 𝐻)
51	1	simprd 497	. . . . 5 ⊢ (𝜑 → ¬ 2 ∥ (𝐺 − 𝐸))
52	51	adantr 482	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → ¬ 2 ∥ (𝐺 − 𝐸))
53	15	breq2d 5161	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (2 ∥ (𝐻 − 𝐹) ↔ 2 ∥ (𝐺 − 𝐸)))
54	52, 53	mtbird 325	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → ¬ 2 ∥ (𝐻 − 𝐹))
55		2z 12594	. . . . . . 7 ⊢ 2 ∈ ℤ
56	9	nn0zd 12584	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℤ)
57	11	nn0zd 12584	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℤ)
58	56, 57	zsubcld 12671	. . . . . . 7 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℤ)
59		dvdsaddr 16246	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ (𝐺 − 𝐸) ∈ ℤ) → (2 ∥ (𝐺 − 𝐸) ↔ 2 ∥ ((𝐺 − 𝐸) + 2)))
60	55, 58, 59	sylancr 588	. . . . . 6 ⊢ (𝜑 → (2 ∥ (𝐺 − 𝐸) ↔ 2 ∥ ((𝐺 − 𝐸) + 2)))
61	51, 60	mtbid 324	. . . . 5 ⊢ (𝜑 → ¬ 2 ∥ ((𝐺 − 𝐸) + 2))
62	61	adantr 482	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ¬ 2 ∥ ((𝐺 − 𝐸) + 2))
63		2cnd 12290	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
64	8, 13, 63	subaddd 11589	. . . . . 6 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 ↔ ((𝐺 − 𝐸) + 2) = (𝐻 − 𝐹)))
65	64	biimpa 478	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ((𝐺 − 𝐸) + 2) = (𝐻 − 𝐹))
66	65	breq2d 5161	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (2 ∥ ((𝐺 − 𝐸) + 2) ↔ 2 ∥ (𝐻 − 𝐹)))
67	62, 66	mtbid 324	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ¬ 2 ∥ (𝐻 − 𝐹))
68	54, 67, 49	mpjaodan 958	. 2 ⊢ (𝜑 → ¬ 2 ∥ (𝐻 − 𝐹))
69	50, 68	jca 513	1 ⊢ (𝜑 → (𝐹 < 𝐻 ∧ ¬ 2 ∥ (𝐻 − 𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 {cpr 4631 class class class wbr 5149 (class class class)co 7409 ℝcr 11109 0cc0 11110 + caddc 11113 < clt 11248 − cmin 11444 2c2 12267 ℕ₀cn0 12472 ℤcz 12558 ∥ cdvds 16197

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-dvds 16198

This theorem is referenced by: (None)

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