signslema - Mathbox for Thierry Arnoux

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Theorem signslema 34820

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 498	. . . . 5 ⊢ (𝜑 → 𝐸 < 𝐺)
3	2	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → 𝐸 < 𝐺)
4		signslema.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ ℕ₀)
5	4	nn0cnd 12541	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ ℂ)
6		signslema.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ℕ₀)
7	6	nn0cnd 12541	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ℂ)
8	5, 7	subcld 11539	. . . . . . . 8 ⊢ (𝜑 → (𝐻 − 𝐹) ∈ ℂ)
9		signslema.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ ℕ₀)
10	9	nn0cnd 12541	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ ℂ)
11		signslema.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
12	11	nn0cnd 12541	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℂ)
13	10, 12	subcld 11539	. . . . . . . 8 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℂ)
14	8, 13	subeq0ad 11549	. . . . . . 7 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0 ↔ (𝐻 − 𝐹) = (𝐺 − 𝐸)))
15	14	biimpa 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐻 − 𝐹) = (𝐺 − 𝐸))
16	15	breq2d 5111	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (0 < (𝐻 − 𝐹) ↔ 0 < (𝐺 − 𝐸)))
17	6	nn0red 12540	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ ℝ)
18	4	nn0red 12540	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ ℝ)
19	17, 18	posdifd 11771	. . . . . 6 ⊢ (𝜑 → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
20	19	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → (𝐹 < 𝐻 ↔ 0 < (𝐻 − 𝐹)))
21	11	nn0red 12540	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
22	9	nn0red 12540	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ ℝ)
23	21, 22	posdifd 11771	. . . . . 6 ⊢ (𝜑 → (𝐸 < 𝐺 ↔ 0 < (𝐺 − 𝐸)))
24	23	adantr 484	. . . . 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 11181	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 ∈ ℝ)
28	22, 21	resubcld 11612	. . . . . 6 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℝ)
29	28	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐺 − 𝐸) ∈ ℝ)
30	18, 17	resubcld 11612	. . . . . 6 ⊢ (𝜑 → (𝐻 − 𝐹) ∈ ℝ)
31	30	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐻 − 𝐹) ∈ ℝ)
32	2	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 𝐸 < 𝐺)
33	23	adantr 484	. . . . . 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 12319	. . . . . . 7 ⊢ 0 < 2
36		breq2 5103	. . . . . . 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 11771	. . . . . . 7 ⊢ (𝜑 → ((𝐺 − 𝐸) < (𝐻 − 𝐹) ↔ 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸))))
39	38	biimpar 481	. . . . . 6 ⊢ ((𝜑 ∧ 0 < ((𝐻 − 𝐹) − (𝐺 − 𝐸))) → (𝐺 − 𝐸) < (𝐻 − 𝐹))
40	37, 39	sylan2 602	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → (𝐺 − 𝐸) < (𝐻 − 𝐹))
41	27, 29, 31, 34, 40	lttrd 11341	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → 0 < (𝐻 − 𝐹))
42	19	adantr 484	. . . 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 11588	. . . . 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 2862	. . . 4 ⊢ (𝜑 → ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ {0, 2})
47		ovex 7425	. . . . 5 ⊢ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) ∈ V
48	47	elpr 4606	. . . 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 971	. 2 ⊢ (𝜑 → 𝐹 < 𝐻)
51	1	simprd 499	. . . . 5 ⊢ (𝜑 → ¬ 2 ∥ (𝐺 − 𝐸))
52	51	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 0) → ¬ 2 ∥ (𝐺 − 𝐸))
53	15	breq2d 5111	. . . 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 12600	. . . . . . 7 ⊢ 2 ∈ ℤ
56	9	nn0zd 12590	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ℤ)
57	11	nn0zd 12590	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℤ)
58	56, 57	zsubcld 12679	. . . . . . 7 ⊢ (𝜑 → (𝐺 − 𝐸) ∈ ℤ)
59		dvdsaddr 16320	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ (𝐺 − 𝐸) ∈ ℤ) → (2 ∥ (𝐺 − 𝐸) ↔ 2 ∥ ((𝐺 − 𝐸) + 2)))
60	55, 58, 59	sylancr 596	. . . . . 6 ⊢ (𝜑 → (2 ∥ (𝐺 − 𝐸) ↔ 2 ∥ ((𝐺 − 𝐸) + 2)))
61	51, 60	mtbid 326	. . . . 5 ⊢ (𝜑 → ¬ 2 ∥ ((𝐺 − 𝐸) + 2))
62	61	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ¬ 2 ∥ ((𝐺 − 𝐸) + 2))
63		2cnd 12293	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℂ)
64	8, 13, 63	subaddd 11557	. . . . . 6 ⊢ (𝜑 → (((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2 ↔ ((𝐺 − 𝐸) + 2) = (𝐻 − 𝐹)))
65	64	biimpa 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝐻 − 𝐹) − (𝐺 − 𝐸)) = 2) → ((𝐺 − 𝐸) + 2) = (𝐻 − 𝐹))
66	65	breq2d 5111	. . . 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 971	. 2 ⊢ (𝜑 → ¬ 2 ∥ (𝐻 − 𝐹))
69	50, 68	jca 519	1 ⊢ (𝜑 → (𝐹 < 𝐻 ∧ ¬ 2 ∥ (𝐻 − 𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 = wceq 1559 ∈ wcel 2141 {cpr 4583 class class class wbr 5099 (class class class)co 7392 ℝcr 11069 0cc0 11070 + caddc 11073 < clt 11213 − cmin 11411 2c2 12269 ℕ₀cn0 12478 ℤcz 12565 ∥ cdvds 16269

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-n0 12479 df-z 12566 df-dvds 16270

This theorem is referenced by: (None)

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