Theorem xmulpnf1 12937

Description: Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xmulpnf1	⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞)

Proof of Theorem xmulpnf1

Step	Hyp	Ref	Expression
1		pnfxr 10960	. . . 4 ⊢ +∞ ∈ ℝ*
2		xmulval 12888	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 ·e +∞) = if((𝐴 = 0 ∨ +∞ = 0), 0, if((((0 < +∞ ∧ 𝐴 = +∞) ∨ (+∞ < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ +∞ = +∞) ∨ (𝐴 < 0 ∧ +∞ = -∞))), +∞, if((((0 < +∞ ∧ 𝐴 = -∞) ∨ (+∞ < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ +∞ = -∞) ∨ (𝐴 < 0 ∧ +∞ = +∞))), -∞, (𝐴 · +∞)))))
3	1, 2	mpan2 687	. . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e +∞) = if((𝐴 = 0 ∨ +∞ = 0), 0, if((((0 < +∞ ∧ 𝐴 = +∞) ∨ (+∞ < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ +∞ = +∞) ∨ (𝐴 < 0 ∧ +∞ = -∞))), +∞, if((((0 < +∞ ∧ 𝐴 = -∞) ∨ (+∞ < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ +∞ = -∞) ∨ (𝐴 < 0 ∧ +∞ = +∞))), -∞, (𝐴 · +∞)))))
4	3	adantr 480	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = if((𝐴 = 0 ∨ +∞ = 0), 0, if((((0 < +∞ ∧ 𝐴 = +∞) ∨ (+∞ < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ +∞ = +∞) ∨ (𝐴 < 0 ∧ +∞ = -∞))), +∞, if((((0 < +∞ ∧ 𝐴 = -∞) ∨ (+∞ < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ +∞ = -∞) ∨ (𝐴 < 0 ∧ +∞ = +∞))), -∞, (𝐴 · +∞)))))
5		0xr 10953	. . . . 5 ⊢ 0 ∈ ℝ*
6		xrltne 12826	. . . . 5 ⊢ ((0 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 𝐴 ≠ 0)
7	5, 6	mp3an1 1446	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 𝐴 ≠ 0)
8		0re 10908	. . . . . 6 ⊢ 0 ∈ ℝ
9		renepnf 10954	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
10	8, 9	ax-mp 5	. . . . 5 ⊢ 0 ≠ +∞
11	10	necomi 2997	. . . 4 ⊢ +∞ ≠ 0
12		neanior 3036	. . . 4 ⊢ ((𝐴 ≠ 0 ∧ +∞ ≠ 0) ↔ ¬ (𝐴 = 0 ∨ +∞ = 0))
13	7, 11, 12	sylanblc 588	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → ¬ (𝐴 = 0 ∨ +∞ = 0))
14	13	iffalsed 4467	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → if((𝐴 = 0 ∨ +∞ = 0), 0, if((((0 < +∞ ∧ 𝐴 = +∞) ∨ (+∞ < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ +∞ = +∞) ∨ (𝐴 < 0 ∧ +∞ = -∞))), +∞, if((((0 < +∞ ∧ 𝐴 = -∞) ∨ (+∞ < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ +∞ = -∞) ∨ (𝐴 < 0 ∧ +∞ = +∞))), -∞, (𝐴 · +∞)))) = if((((0 < +∞ ∧ 𝐴 = +∞) ∨ (+∞ < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ +∞ = +∞) ∨ (𝐴 < 0 ∧ +∞ = -∞))), +∞, if((((0 < +∞ ∧ 𝐴 = -∞) ∨ (+∞ < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ +∞ = -∞) ∨ (𝐴 < 0 ∧ +∞ = +∞))), -∞, (𝐴 · +∞))))
15		simpr 484	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → 0 < 𝐴)
16		eqid 2738	. . . . . 6 ⊢ +∞ = +∞
17	15, 16	jctir 520	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (0 < 𝐴 ∧ +∞ = +∞))
18	17	orcd 869	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → ((0 < 𝐴 ∧ +∞ = +∞) ∨ (𝐴 < 0 ∧ +∞ = -∞)))
19	18	olcd 870	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (((0 < +∞ ∧ 𝐴 = +∞) ∨ (+∞ < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ +∞ = +∞) ∨ (𝐴 < 0 ∧ +∞ = -∞))))
20	19	iftrued 4464	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → if((((0 < +∞ ∧ 𝐴 = +∞) ∨ (+∞ < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ +∞ = +∞) ∨ (𝐴 < 0 ∧ +∞ = -∞))), +∞, if((((0 < +∞ ∧ 𝐴 = -∞) ∨ (+∞ < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ +∞ = -∞) ∨ (𝐴 < 0 ∧ +∞ = +∞))), -∞, (𝐴 · +∞))) = +∞)
21	4, 14, 20	3eqtrd 2782	1 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ifcif 4456   class class class wbr 5070  (class class class)co 7255  ℝcr 10801  0cc0 10802   · cmul 10807  +∞cpnf 10937  -∞cmnf 10938  ℝ^*cxr 10939   < clt 10940   ·_e cxmu 12776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-i2m1 10870  ax-rnegex 10873  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-xmul 12779

This theorem is referenced by:  xmulpnf2  12938  xmulmnf1  12939  xmulpnf1n  12941  xmulgt0  12946  xmulasslem3  12949  xlemul1a  12951  xadddilem  12957  nn0xmulclb  30996  hashxpe  31029  xdivpnfrp  31109  xrge0adddir  31203  esumcst  31931  esumpinfval  31941