signslema - Mathbox for Thierry Arnoux

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Theorem signslema 31834

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 497	. . . . 5 ⊢ (𝜑 → 𝐸 < 𝐺)
3	2	adantr 483	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → 𝐸 < 𝐺)
4		signslema.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
5	4	nn0cnd 11960	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ ℂ)
6		signslema.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ℕ₀)
7	6	nn0cnd 11960	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ℂ)
8	5, 7	subcld 10999	. . . . . . . 8 ⊢ (𝜑 → (𝐻 − 𝐹) ∈ ℂ)
9		signslema.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ ℕ₀)
10	9	nn0cnd 11960	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ ℂ)
11		signslema.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
12	11	nn0cnd 11960	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℂ)
13	10, 12	subcld 10999	. . . . . . . 8 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℂ)
14	8, 13	subeq0ad 11009	. . . . . . 7 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ↔ (𝐻 − 𝐹) = (𝐺 − 𝐸)))
15	14	biimpa 479	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐻 − 𝐹) = (𝐺 − 𝐸))
16	15	breq2d 5080	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (0 < (𝐻 − 𝐹) ↔ 0 < (𝐺 − 𝐸)))
17	6	nn0red 11959	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ℝ)
18	4	nn0red 11959	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ ℝ)
19	17, 18	posdifd 11229	. . . . . 6 ⊢ (𝜑 → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
20	19	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
21	11	nn0red 11959	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
22	9	nn0red 11959	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ ℝ)
23	21, 22	posdifd 11229	. . . . . 6 ⊢ (𝜑 → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
24	23	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
25	16, 20, 24	3bitr4rd 314	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐸 < 𝐺 ↔ 𝐹 < 𝐻))
26	3, 25	mpbid 234	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → 𝐹 < 𝐻)
27		0red 10646	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 ∈ ℝ)
28	22, 21	resubcld 11070	. . . . . 6 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℝ)
29	28	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐺 − 𝐸) ∈ ℝ)
30	18, 17	resubcld 11070	. . . . . 6 ⊢ (𝜑 → (𝐻 − 𝐹) ∈ ℝ)
31	30	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐻 − 𝐹) ∈ ℝ)
32	2	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 𝐸 < 𝐺)
33	23	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
34	32, 33	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 < (𝐺 − 𝐸))
35		2pos 11743	. . . . . . 7 ⊢ 0 < 2
36		breq2 5072	. . . . . . 7 ⊢ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 → (0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ↔ 0 < 2))
37	35, 36	mpbiri 260	. . . . . 6 ⊢ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 → 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸)))
38	28, 30	posdifd 11229	. . . . . . 7 ⊢ (𝜑 → ((𝐺 − 𝐸) < (𝐻 − 𝐹) ↔ 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸))))
39	38	biimpar 480	. . . . . 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 10803	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 < (𝐻 − 𝐹))
42	19	adantr 483	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
43	41, 42	mpbird 259	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 𝐹 < 𝐻)
44	5, 10, 7, 12	sub4d 11048	. . . . 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 2916	. . . 4 ⊢ (𝜑 → ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ {0, 2})
47		ovex 7191	. . . . 5 ⊢ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ V
48	47	elpr 4592	. . . 4 ⊢ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ {0, 2} ↔ (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ∨ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2))
49	46, 48	sylib 220	. . 3 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ∨ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2))
50	26, 43, 49	mpjaodan 955	. 2 ⊢ (𝜑 → 𝐹 < 𝐻)
51	1	simprd 498	. . . . 5 ⊢ (𝜑 → ¬ 2 ∥ (𝐺 − 𝐸))
52	51	adantr 483	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → ¬ 2 ∥ (𝐺 − 𝐸))
53	15	breq2d 5080	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (2 ∥ (𝐻 − 𝐹) ↔ 2 ∥ (𝐺 − 𝐸)))
54	52, 53	mtbird 327	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → ¬ 2 ∥ (𝐻 − 𝐹))
55		2z 12017	. . . . . . 7 ⊢ 2 ∈ ℤ
56	9	nn0zd 12088	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℤ)
57	11	nn0zd 12088	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℤ)
58	56, 57	zsubcld 12095	. . . . . . 7 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℤ)
59		dvdsaddr 15655	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ (𝐺 − 𝐸) ∈ ℤ) → (2 ∥ (𝐺 − 𝐸) ↔ 2 ∥ ((𝐺 − 𝐸) + 2)))
60	55, 58, 59	sylancr 589	. . . . . 6 ⊢ (𝜑 → (2 ∥ (𝐺 − 𝐸) ↔ 2 ∥ ((𝐺 − 𝐸) + 2)))
61	51, 60	mtbid 326	. . . . 5 ⊢ (𝜑 → ¬ 2 ∥ ((𝐺 − 𝐸) + 2))
62	61	adantr 483	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ¬ 2 ∥ ((𝐺 − 𝐸) + 2))
63		2cnd 11718	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
64	8, 13, 63	subaddd 11017	. . . . . 6 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 ↔ ((𝐺 − 𝐸) + 2) = (𝐻 − 𝐹)))
65	64	biimpa 479	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ((𝐺 − 𝐸) + 2) = (𝐻 − 𝐹))
66	65	breq2d 5080	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (2 ∥ ((𝐺 − 𝐸) + 2) ↔ 2 ∥ (𝐻 − 𝐹)))
67	62, 66	mtbid 326	. . 3 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ¬ 2 ∥ (𝐻 − 𝐹))
68	54, 67, 49	mpjaodan 955	. 2 ⊢ (𝜑 → ¬ 2 ∥ (𝐻 − 𝐹))
69	50, 68	jca 514	1 ⊢ (𝜑 → (𝐹 < 𝐻 ∧ ¬ 2 ∥ (𝐻 − 𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {cpr 4571 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 0cc0 10539 + caddc 10542 < clt 10677 − cmin 10872 2c2 11695 ℕ₀cn0 11900 ℤcz 11984 ∥ cdvds 15609

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-dvds 15610

This theorem is referenced by: (None)

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