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Theorem algcvgblem 12739

Description: Lemma for algcvgb 12740. (Contributed by Paul Chapman, 31-Mar-2011.)

**Assertion**
Ref	Expression
algcvgblem	⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑁 ≠ 0 → 𝑁 < 𝑀) ↔ ((𝑀 ≠ 0 → 𝑁 < 𝑀) ∧ (𝑀 = 0 → 𝑁 = 0))))

**Proof of Theorem algcvgblem**
Step	Hyp	Ref	Expression
1		nn0z 9593	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
2		0z 9584	. . . . . . . . 9 ⊢ 0 ∈ ℤ
3		zdceq 9649	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
4	1, 2, 3	sylancl 413	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → DECID 𝑁 = 0)
5	4	dcned 2418	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → DECID 𝑁 ≠ 0)
6		imordc 905	. . . . . . 7 ⊢ (DECID 𝑁 ≠ 0 → ((𝑁 ≠ 0 → 𝑁 < 𝑀) ↔ (¬ 𝑁 ≠ 0 ∨ 𝑁 < 𝑀)))
7	5, 6	syl 14	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 ≠ 0 → 𝑁 < 𝑀) ↔ (¬ 𝑁 ≠ 0 ∨ 𝑁 < 𝑀)))
8	7	adantl 277	. . . . 5 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑁 ≠ 0 → 𝑁 < 𝑀) ↔ (¬ 𝑁 ≠ 0 ∨ 𝑁 < 𝑀)))
9		nn0z 9593	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
10		zltnle 9619	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (0 < 𝑀 ↔ ¬ 𝑀 ≤ 0))
11	2, 9, 10	sylancr 414	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ₀ → (0 < 𝑀 ↔ ¬ 𝑀 ≤ 0))
12	11	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0 < 𝑀 ↔ ¬ 𝑀 ≤ 0))
13		nn0le0eq0 9520	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 ≤ 0 ↔ 𝑀 = 0))
14	13	notbid 673	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ₀ → (¬ 𝑀 ≤ 0 ↔ ¬ 𝑀 = 0))
15	14	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑀 ≤ 0 ↔ ¬ 𝑀 = 0))
16	12, 15	bitrd 188	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0 < 𝑀 ↔ ¬ 𝑀 = 0))
17		df-ne 2413	. . . . . . . . . . 11 ⊢ (𝑀 ≠ 0 ↔ ¬ 𝑀 = 0)
18	16, 17	bitr4di 198	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (0 < 𝑀 ↔ 𝑀 ≠ 0))
19	18	anbi2d 464	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((¬ 𝑁 ≠ 0 ∧ 0 < 𝑀) ↔ (¬ 𝑁 ≠ 0 ∧ 𝑀 ≠ 0)))
20	1	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℤ)
21	20, 2, 3	sylancl 413	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → DECID 𝑁 = 0)
22		nnedc 2417	. . . . . . . . . . . . 13 ⊢ (DECID 𝑁 = 0 → (¬ 𝑁 ≠ 0 ↔ 𝑁 = 0))
23	21, 22	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑁 ≠ 0 ↔ 𝑁 = 0))
24		breq1 4111	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 → (𝑁 < 𝑀 ↔ 0 < 𝑀))
25	23, 24	biimtrdi 163	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑁 ≠ 0 → (𝑁 < 𝑀 ↔ 0 < 𝑀)))
26		biimpr 130	. . . . . . . . . . 11 ⊢ ((𝑁 < 𝑀 ↔ 0 < 𝑀) → (0 < 𝑀 → 𝑁 < 𝑀))
27	25, 26	syl6 33	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑁 ≠ 0 → (0 < 𝑀 → 𝑁 < 𝑀)))
28	27	impd 254	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((¬ 𝑁 ≠ 0 ∧ 0 < 𝑀) → 𝑁 < 𝑀))
29	19, 28	sylbird 170	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((¬ 𝑁 ≠ 0 ∧ 𝑀 ≠ 0) → 𝑁 < 𝑀))
30	29	expd 258	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑁 ≠ 0 → (𝑀 ≠ 0 → 𝑁 < 𝑀)))
31		ax-1 6	. . . . . . 7 ⊢ (𝑁 < 𝑀 → (𝑀 ≠ 0 → 𝑁 < 𝑀))
32	30, 31	jctir 313	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((¬ 𝑁 ≠ 0 → (𝑀 ≠ 0 → 𝑁 < 𝑀)) ∧ (𝑁 < 𝑀 → (𝑀 ≠ 0 → 𝑁 < 𝑀))))
33		jaob 718	. . . . . 6 ⊢ (((¬ 𝑁 ≠ 0 ∨ 𝑁 < 𝑀) → (𝑀 ≠ 0 → 𝑁 < 𝑀)) ↔ ((¬ 𝑁 ≠ 0 → (𝑀 ≠ 0 → 𝑁 < 𝑀)) ∧ (𝑁 < 𝑀 → (𝑀 ≠ 0 → 𝑁 < 𝑀))))
34	32, 33	sylibr 134	. . . . 5 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((¬ 𝑁 ≠ 0 ∨ 𝑁 < 𝑀) → (𝑀 ≠ 0 → 𝑁 < 𝑀)))
35	8, 34	sylbid 150	. . . 4 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑁 ≠ 0 → 𝑁 < 𝑀) → (𝑀 ≠ 0 → 𝑁 < 𝑀)))
36		nn0ge0 9517	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
37	36	adantl 277	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 ≤ 𝑁)
38		nn0re 9501	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
39		nn0re 9501	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
40		0re 8270	. . . . . . . . 9 ⊢ 0 ∈ ℝ
41		lelttr 8358	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((0 ≤ 𝑁 ∧ 𝑁 < 𝑀) → 0 < 𝑀))
42	40, 41	mp3an1 1361	. . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((0 ≤ 𝑁 ∧ 𝑁 < 𝑀) → 0 < 𝑀))
43	38, 39, 42	syl2anr 290	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0 ≤ 𝑁 ∧ 𝑁 < 𝑀) → 0 < 𝑀))
44	37, 43	mpand 429	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 < 𝑀 → 0 < 𝑀))
45	44, 18	sylibd 149	. . . . 5 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 < 𝑀 → 𝑀 ≠ 0))
46	45	imim2d 54	. . . 4 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑁 ≠ 0 → 𝑁 < 𝑀) → (𝑁 ≠ 0 → 𝑀 ≠ 0)))
47	35, 46	jcad 307	. . 3 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑁 ≠ 0 → 𝑁 < 𝑀) → ((𝑀 ≠ 0 → 𝑁 < 𝑀) ∧ (𝑁 ≠ 0 → 𝑀 ≠ 0))))
48		pm3.34 346	. . 3 ⊢ (((𝑀 ≠ 0 → 𝑁 < 𝑀) ∧ (𝑁 ≠ 0 → 𝑀 ≠ 0)) → (𝑁 ≠ 0 → 𝑁 < 𝑀))
49	47, 48	impbid1 142	. 2 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑁 ≠ 0 → 𝑁 < 𝑀) ↔ ((𝑀 ≠ 0 → 𝑁 < 𝑀) ∧ (𝑁 ≠ 0 → 𝑀 ≠ 0))))
50		con34bdc 879	. . . . 5 ⊢ (DECID 𝑁 = 0 → ((𝑀 = 0 → 𝑁 = 0) ↔ (¬ 𝑁 = 0 → ¬ 𝑀 = 0)))
51	21, 50	syl 14	. . . 4 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑀 = 0 → 𝑁 = 0) ↔ (¬ 𝑁 = 0 → ¬ 𝑀 = 0)))
52		df-ne 2413	. . . . 5 ⊢ (𝑁 ≠ 0 ↔ ¬ 𝑁 = 0)
53	52, 17	imbi12i 239	. . . 4 ⊢ ((𝑁 ≠ 0 → 𝑀 ≠ 0) ↔ (¬ 𝑁 = 0 → ¬ 𝑀 = 0))
54	51, 53	bitr4di 198	. . 3 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑀 = 0 → 𝑁 = 0) ↔ (𝑁 ≠ 0 → 𝑀 ≠ 0)))
55	54	anbi2d 464	. 2 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (((𝑀 ≠ 0 → 𝑁 < 𝑀) ∧ (𝑀 = 0 → 𝑁 = 0)) ↔ ((𝑀 ≠ 0 → 𝑁 < 𝑀) ∧ (𝑁 ≠ 0 → 𝑀 ≠ 0))))
56	49, 55	bitr4d 191	1 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑁 ≠ 0 → 𝑁 < 𝑀) ↔ ((𝑀 ≠ 0 → 𝑁 < 𝑀) ∧ (𝑀 = 0 → 𝑁 = 0))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 = wceq 1398 ∈ wcel 2203 ≠ wne 2412 class class class wbr 4108 ℝcr 8122 0cc0 8123 < clt 8304 ≤ cle 8305 ℕ₀cn0 9492 ℤcz 9573

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4227 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-addcom 8223 ax-addass 8225 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-0id 8231 ax-rnegex 8232 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2814 df-sbc 3042 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-br 4109 df-opab 4171 df-id 4413 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-iota 5311 df-fun 5353 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-inn 9234 df-n0 9493 df-z 9574

This theorem is referenced by: algcvgb 12740

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