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Theorem algcvgblem 12579

Description: Lemma for algcvgb 12580. (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 9474	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
2		0z 9465	. . . . . . . . 9 ⊢ 0 ∈ ℤ
3		zdceq 9530	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
4	1, 2, 3	sylancl 413	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → DECID 𝑁 = 0)
5	4	dcned 2406	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → DECID 𝑁 ≠ 0)
6		imordc 902	. . . . . . 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 9474	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
10		zltnle 9500	. . . . . . . . . . . . . 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 9405	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 ≤ 0 ↔ 𝑀 = 0))
14	13	notbid 671	. . . . . . . . . . . . 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 2401	. . . . . . . . . . 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 2405	. . . . . . . . . . . . 13 ⊢ (DECID 𝑁 = 0 → (¬ 𝑁 ≠ 0 ↔ 𝑁 = 0))
23	21, 22	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑁 ≠ 0 ↔ 𝑁 = 0))
24		breq1 4086	. . . . . . . . . . . 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 715	. . . . . 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 9402	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
37	36	adantl 277	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 ≤ 𝑁)
38		nn0re 9386	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
39		nn0re 9386	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
40		0re 8154	. . . . . . . . 9 ⊢ 0 ∈ ℝ
41		lelttr 8243	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((0 ≤ 𝑁 ∧ 𝑁 < 𝑀) → 0 < 𝑀))
42	40, 41	mp3an1 1358	. . . . . . . 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 876	. . . . 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 2401	. . . . 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 713 DECID wdc 839 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 class class class wbr 4083 ℝcr 8006 0cc0 8007 < clt 8189 ≤ cle 8190 ℕ₀cn0 9377 ℤcz 9454

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-addcom 8107 ax-addass 8109 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-0id 8115 ax-rnegex 8116 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-inn 9119 df-n0 9378 df-z 9455

This theorem is referenced by: algcvgb 12580

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