Theorem expge1 10937

Description: A real greater than or equal to 1 raised to a nonnegative integer is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)

Assertion

Ref	Expression
expge1	⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝐴) → 1 ≤ (𝐴↑𝑁))

Proof of Theorem expge1

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4112	. . . . . 6 ⊢ (𝑧 = 𝐴 → (1 ≤ 𝑧 ↔ 1 ≤ 𝐴))
2	1	elrab 2972	. . . . 5 ⊢ (𝐴 ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} ↔ (𝐴 ∈ ℝ ∧ 1 ≤ 𝐴))
3		ssrab2 3322	. . . . . . 7 ⊢ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} ⊆ ℝ
4		ax-resscn 8218	. . . . . . 7 ⊢ ℝ ⊆ ℂ
5	3, 4	sstri 3246	. . . . . 6 ⊢ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} ⊆ ℂ
6		breq2 4112	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (1 ≤ 𝑧 ↔ 1 ≤ 𝑥))
7	6	elrab 2972	. . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥))
8		breq2 4112	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (1 ≤ 𝑧 ↔ 1 ≤ 𝑦))
9	8	elrab 2972	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} ↔ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦))
10		remulcl 8254	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
11	10	ad2ant2r 509	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (𝑥 · 𝑦) ∈ ℝ)
12		1t1e1 9389	. . . . . . . . . 10 ⊢ (1 · 1) = 1
13		1re 8272	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℝ
14		0le1 8754	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 1
15	13, 14	pm3.2i 272	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℝ ∧ 0 ≤ 1)
16	15	jctl 314	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → ((1 ∈ ℝ ∧ 0 ≤ 1) ∧ 𝑥 ∈ ℝ))
17	15	jctl 314	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → ((1 ∈ ℝ ∧ 0 ≤ 1) ∧ 𝑦 ∈ ℝ))
18		lemul12a 9135	. . . . . . . . . . . 12 ⊢ ((((1 ∈ ℝ ∧ 0 ≤ 1) ∧ 𝑥 ∈ ℝ) ∧ ((1 ∈ ℝ ∧ 0 ≤ 1) ∧ 𝑦 ∈ ℝ)) → ((1 ≤ 𝑥 ∧ 1 ≤ 𝑦) → (1 · 1) ≤ (𝑥 · 𝑦)))
19	16, 17, 18	syl2an 289	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((1 ≤ 𝑥 ∧ 1 ≤ 𝑦) → (1 · 1) ≤ (𝑥 · 𝑦)))
20	19	imp 124	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (1 ≤ 𝑥 ∧ 1 ≤ 𝑦)) → (1 · 1) ≤ (𝑥 · 𝑦))
21	12, 20	eqbrtrrid 4144	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (1 ≤ 𝑥 ∧ 1 ≤ 𝑦)) → 1 ≤ (𝑥 · 𝑦))
22	21	an4s 592	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ≤ (𝑥 · 𝑦))
23		breq2 4112	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 · 𝑦) → (1 ≤ 𝑧 ↔ 1 ≤ (𝑥 · 𝑦)))
24	23	elrab 2972	. . . . . . . 8 ⊢ ((𝑥 · 𝑦) ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} ↔ ((𝑥 · 𝑦) ∈ ℝ ∧ 1 ≤ (𝑥 · 𝑦)))
25	11, 22, 24	sylanbrc 417	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (𝑥 · 𝑦) ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧})
26	7, 9, 25	syl2anb 291	. . . . . 6 ⊢ ((𝑥 ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧}) → (𝑥 · 𝑦) ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧})
27		1le1 8845	. . . . . . 7 ⊢ 1 ≤ 1
28		breq2 4112	. . . . . . . 8 ⊢ (𝑧 = 1 → (1 ≤ 𝑧 ↔ 1 ≤ 1))
29	28	elrab 2972	. . . . . . 7 ⊢ (1 ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} ↔ (1 ∈ ℝ ∧ 1 ≤ 1))
30	13, 27, 29	mpbir2an 951	. . . . . 6 ⊢ 1 ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧}
31	5, 26, 30	expcllem 10911	. . . . 5 ⊢ ((𝐴 ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧})
32	2, 31	sylanbr 285	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧})
33	32	3impa 1221	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧})
34	33	3com23 1236	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝐴) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧})
35		breq2 4112	. . . 4 ⊢ (𝑧 = (𝐴↑𝑁) → (1 ≤ 𝑧 ↔ 1 ≤ (𝐴↑𝑁)))
36	35	elrab 2972	. . 3 ⊢ ((𝐴↑𝑁) ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} ↔ ((𝐴↑𝑁) ∈ ℝ ∧ 1 ≤ (𝐴↑𝑁)))
37	36	simprbi 275	. 2 ⊢ ((𝐴↑𝑁) ∈ {𝑧 ∈ ℝ ∣ 1 ≤ 𝑧} → 1 ≤ (𝐴↑𝑁))
38	34, 37	syl 14	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝐴) → 1 ≤ (𝐴↑𝑁))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   ∈ wcel 2203  {crab 2524   class class class wbr 4108  (class class class)co 6049  ℂcc 8124  ℝcr 8125  0cc0 8126  1c1 8127   · cmul 8131   ≤ cle 8308  ℕ₀cn0 9495  ↑cexp 10899

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-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244

This theorem depends on definitions:  df-bi 117  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-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-n0 9496  df-z 9577  df-uz 9853  df-seqfrec 10809  df-exp 10900

This theorem is referenced by:  expgt1  10938  leexp2a  10953  expge1d  11053