rexmul - Metamath Proof Explorer

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Theorem rexmul 13231

Description: The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
rexmul	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))

**Proof of Theorem rexmul**
Step	Hyp	Ref	Expression
1		renepnf 11222	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
2	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≠ +∞)
3	2	necon2bi 2955	. . . . . . . . 9 ⊢ (𝐴 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
4	3	adantl 481	. . . . . . . 8 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
5		renemnf 11223	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
6	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≠ -∞)
7	6	necon2bi 2955	. . . . . . . . 9 ⊢ (𝐴 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
8	7	adantl 481	. . . . . . . 8 ⊢ ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
9	4, 8	jaoi 857	. . . . . . 7 ⊢ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
10		renepnf 11222	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
11	10	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≠ +∞)
12	11	necon2bi 2955	. . . . . . . . 9 ⊢ (𝐵 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
13	12	adantl 481	. . . . . . . 8 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
14		renemnf 11223	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
15	14	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≠ -∞)
16	15	necon2bi 2955	. . . . . . . . 9 ⊢ (𝐵 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
17	16	adantl 481	. . . . . . . 8 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
18	13, 17	jaoi 857	. . . . . . 7 ⊢ (((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
19	9, 18	jaoi 857	. . . . . 6 ⊢ ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
20	19	con2i 139	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
21	20	iffalsed 4499	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))
22	7	adantl 481	. . . . . . . 8 ⊢ ((0 < 𝐵 ∧ 𝐴 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
23	3	adantl 481	. . . . . . . 8 ⊢ ((𝐵 < 0 ∧ 𝐴 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
24	22, 23	jaoi 857	. . . . . . 7 ⊢ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
25	16	adantl 481	. . . . . . . 8 ⊢ ((0 < 𝐴 ∧ 𝐵 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
26	12	adantl 481	. . . . . . . 8 ⊢ ((𝐴 < 0 ∧ 𝐵 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
27	25, 26	jaoi 857	. . . . . . 7 ⊢ (((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
28	24, 27	jaoi 857	. . . . . 6 ⊢ ((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
29	28	con2i 139	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
30	29	iffalsed 4499	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)) = (𝐴 · 𝐵))
31	21, 30	eqtrd 2764	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵))) = (𝐴 · 𝐵))
32	31	ifeq2d 4509	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, (𝐴 · 𝐵)))
33		rexr 11220	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
34		rexr 11220	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
35		xmulval 13185	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ·_e 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
36	33, 34, 35	syl2an 596	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·_e 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, if((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))), +∞, if((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))), -∞, (𝐴 · 𝐵)))))
37		ifid 4529	. . 3 ⊢ if((𝐴 = 0 ∨ 𝐵 = 0), (𝐴 · 𝐵), (𝐴 · 𝐵)) = (𝐴 · 𝐵)
38		oveq1 7394	. . . . . 6 ⊢ (𝐴 = 0 → (𝐴 · 𝐵) = (0 · 𝐵))
39		mul02lem2 11351	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (0 · 𝐵) = 0)
40	39	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 · 𝐵) = 0)
41	38, 40	sylan9eqr 2786	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → (𝐴 · 𝐵) = 0)
42		oveq2 7395	. . . . . 6 ⊢ (𝐵 = 0 → (𝐴 · 𝐵) = (𝐴 · 0))
43		recn 11158	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
44	43	mul01d 11373	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0)
45	44	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 0) = 0)
46	42, 45	sylan9eqr 2786	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐵 = 0) → (𝐴 · 𝐵) = 0)
47	41, 46	jaodan 959	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 = 0 ∨ 𝐵 = 0)) → (𝐴 · 𝐵) = 0)
48	47	ifeq1da 4520	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if((𝐴 = 0 ∨ 𝐵 = 0), (𝐴 · 𝐵), (𝐴 · 𝐵)) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, (𝐴 · 𝐵)))
49	37, 48	eqtr3id 2778	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, (𝐴 · 𝐵)))
50	32, 36, 49	3eqtr4d 2774	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ifcif 4488 class class class wbr 5107 (class class class)co 7387 ℝcr 11067 0cc0 11068 · cmul 11073 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ·_e cxmu 13071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-po 5546 df-so 5547 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-xmul 13074

This theorem is referenced by: xmulrid 13239 xmulgt0 13243 xmulasslem3 13246 xlemul1a 13248 xlemul1 13250 xadddilem 13254 nmoix 24617 nmoi2 24618 metnrmlem3 24750 nmoleub2lem 25014 nn0xmulclb 32694 hashxpe 32732 xrecex 32840 rexdiv 32846 pnfinf 33137 xrge0slmod 33319 fldextrspundgdvdslem 33675 fldextrspundgdvds 33676 fldext2rspun 33677 fldext2chn 33718 constrext2chnlem 33740 esumcst 34053 omssubadd 34291

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