signslema - Mathbox for Thierry Arnoux

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Theorem signslema 34722

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 494	. . . . 5 ⊢ (𝜑 → 𝐸 < 𝐺)
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → 𝐸 < 𝐺)
4		signslema.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
5	4	nn0cnd 12491	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ ℂ)
6		signslema.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ℕ₀)
7	6	nn0cnd 12491	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ℂ)
8	5, 7	subcld 11496	. . . . . . . 8 ⊢ (𝜑 → (𝐻 − 𝐹) ∈ ℂ)
9		signslema.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ ℕ₀)
10	9	nn0cnd 12491	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ ℂ)
11		signslema.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
12	11	nn0cnd 12491	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℂ)
13	10, 12	subcld 11496	. . . . . . . 8 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℂ)
14	8, 13	subeq0ad 11506	. . . . . . 7 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ↔ (𝐻 − 𝐹) = (𝐺 − 𝐸)))
15	14	biimpa 476	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐻 − 𝐹) = (𝐺 − 𝐸))
16	15	breq2d 5098	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (0 < (𝐻 − 𝐹) ↔ 0 < (𝐺 − 𝐸)))
17	6	nn0red 12490	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ℝ)
18	4	nn0red 12490	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ ℝ)
19	17, 18	posdifd 11728	. . . . . 6 ⊢ (𝜑 → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
21	11	nn0red 12490	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
22	9	nn0red 12490	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ ℝ)
23	21, 22	posdifd 11728	. . . . . 6 ⊢ (𝜑 → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
24	23	adantr 480	. . . . 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 232	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → 𝐹 < 𝐻)
27		0red 11138	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 ∈ ℝ)
28	22, 21	resubcld 11569	. . . . . 6 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℝ)
29	28	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐺 − 𝐸) ∈ ℝ)
30	18, 17	resubcld 11569	. . . . . 6 ⊢ (𝜑 → (𝐻 − 𝐹) ∈ ℝ)
31	30	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐻 − 𝐹) ∈ ℝ)
32	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 𝐸 < 𝐺)
33	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
34	32, 33	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 < (𝐺 − 𝐸))
35		2pos 12275	. . . . . . 7 ⊢ 0 < 2
36		breq2 5090	. . . . . . 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 11728	. . . . . . 7 ⊢ (𝜑 → ((𝐺 − 𝐸) < (𝐻 − 𝐹) ↔ 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸))))
39	38	biimpar 477	. . . . . 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 11298	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 < (𝐻 − 𝐹))
42	19	adantr 480	. . . 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 11545	. . . . 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 2838	. . . 4 ⊢ (𝜑 → ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ {0, 2})
47		ovex 7393	. . . . 5 ⊢ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ V
48	47	elpr 4593	. . . 4 ⊢ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ {0, 2} ↔ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ∨ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2))
49	46, 48	sylib 218	. . 3 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ∨ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2))
50	26, 43, 49	mpjaodan 961	. 2 ⊢ (𝜑 → 𝐹 < 𝐻)
51	1	simprd 495	. . . . 5 ⊢ (𝜑 → ¬ 2 ∥ (𝐺 − 𝐸))
52	51	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → ¬ 2 ∥ (𝐺 − 𝐸))
53	15	breq2d 5098	. . . 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 12550	. . . . . . 7 ⊢ 2 ∈ ℤ
56	9	nn0zd 12540	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℤ)
57	11	nn0zd 12540	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℤ)
58	56, 57	zsubcld 12629	. . . . . . 7 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℤ)
59		dvdsaddr 16263	. . . . . . 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 480	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ¬ 2 ∥ ((𝐺 − 𝐸) + 2))
63		2cnd 12250	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
64	8, 13, 63	subaddd 11514	. . . . . 6 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 ↔ ((𝐺 − 𝐸) + 2) = (𝐻 − 𝐹)))
65	64	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ((𝐺 − 𝐸) + 2) = (𝐻 − 𝐹))
66	65	breq2d 5098	. . . 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 961	. 2 ⊢ (𝜑 → ¬ 2 ∥ (𝐻 − 𝐹))
69	50, 68	jca 511	1 ⊢ (𝜑 → (𝐹 < 𝐻 ∧ ¬ 2 ∥ (𝐻 − 𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 {cpr 4570 class class class wbr 5086 (class class class)co 7360 ℝcr 11028 0cc0 11029 + caddc 11032 < clt 11170 − cmin 11368 2c2 12227 ℕ₀cn0 12428 ℤcz 12515 ∥ cdvds 16212

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 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

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 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-n0 12429 df-z 12516 df-dvds 16213

This theorem is referenced by: (None)

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