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Theorem algcvgblem 12217

Description: Lemma for algcvgb 12218. (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 9346	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
2		0z 9337	. . . . . . . . 9 ⊢ 0 ∈ ℤ
3		zdceq 9401	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
4	1, 2, 3	sylancl 413	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → DECID 𝑁 = 0)
5	4	dcned 2373	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → DECID 𝑁 ≠ 0)
6		imordc 898	. . . . . . 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 9346	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
10		zltnle 9372	. . . . . . . . . . . . . 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 9277	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 ≤ 0 ↔ 𝑀 = 0))
14	13	notbid 668	. . . . . . . . . . . . 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 2368	. . . . . . . . . . 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 2372	. . . . . . . . . . . . 13 ⊢ (DECID 𝑁 = 0 → (¬ 𝑁 ≠ 0 ↔ 𝑁 = 0))
23	21, 22	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑁 ≠ 0 ↔ 𝑁 = 0))
24		breq1 4036	. . . . . . . . . . . 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 711	. . . . . 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 9274	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
37	36	adantl 277	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 0 ≤ 𝑁)
38		nn0re 9258	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
39		nn0re 9258	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
40		0re 8026	. . . . . . . . 9 ⊢ 0 ∈ ℝ
41		lelttr 8115	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((0 ≤ 𝑁 ∧ 𝑁 < 𝑀) → 0 < 𝑀))
42	40, 41	mp3an1 1335	. . . . . . . 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 872	. . . . 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 2368	. . . . 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 709 DECID wdc 835 = wceq 1364 ∈ wcel 2167 ≠ wne 2367 class class class wbr 4033 ℝcr 7878 0cc0 7879 < clt 8061 ≤ cle 8062 ℕ₀cn0 9249 ℤcz 9326

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-addcom 7979 ax-addass 7981 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-0id 7987 ax-rnegex 7988 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-br 4034 df-opab 4095 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-iota 5219 df-fun 5260 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-inn 8991 df-n0 9250 df-z 9327

This theorem is referenced by: algcvgb 12218

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