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Theorem rexmul 13250

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 11262	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
2	1	adantr 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≠ +∞)
3	2	necon2bi 2972	. . . . . . . . 9 ⊢ (𝐴 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
4	3	adantl 483	. . . . . . . 8 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
5		renemnf 11263	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
6	5	adantr 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≠ -∞)
7	6	necon2bi 2972	. . . . . . . . 9 ⊢ (𝐴 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
8	7	adantl 483	. . . . . . . 8 ⊢ ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
9	4, 8	jaoi 856	. . . . . . 7 ⊢ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
10		renepnf 11262	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
11	10	adantl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≠ +∞)
12	11	necon2bi 2972	. . . . . . . . 9 ⊢ (𝐵 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
13	12	adantl 483	. . . . . . . 8 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
14		renemnf 11263	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
15	14	adantl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≠ -∞)
16	15	necon2bi 2972	. . . . . . . . 9 ⊢ (𝐵 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
17	16	adantl 483	. . . . . . . 8 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
18	13, 17	jaoi 856	. . . . . . 7 ⊢ (((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
19	9, 18	jaoi 856	. . . . . 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 4540	. . . 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 483	. . . . . . . 8 ⊢ ((0 < 𝐵 ∧ 𝐴 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
23	3	adantl 483	. . . . . . . 8 ⊢ ((𝐵 < 0 ∧ 𝐴 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
24	22, 23	jaoi 856	. . . . . . 7 ⊢ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
25	16	adantl 483	. . . . . . . 8 ⊢ ((0 < 𝐴 ∧ 𝐵 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
26	12	adantl 483	. . . . . . . 8 ⊢ ((𝐴 < 0 ∧ 𝐵 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
27	25, 26	jaoi 856	. . . . . . 7 ⊢ (((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
28	24, 27	jaoi 856	. . . . . 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 4540	. . . 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 2773	. . 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 4549	. 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 11260	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
34		rexr 11260	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
35		xmulval 13204	. . 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 597	. 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 4569	. . 3 ⊢ if((𝐴 = 0 ∨ 𝐵 = 0), (𝐴 · 𝐵), (𝐴 · 𝐵)) = (𝐴 · 𝐵)
38		oveq1 7416	. . . . . 6 ⊢ (𝐴 = 0 → (𝐴 · 𝐵) = (0 · 𝐵))
39		mul02lem2 11391	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (0 · 𝐵) = 0)
40	39	adantl 483	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 · 𝐵) = 0)
41	38, 40	sylan9eqr 2795	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → (𝐴 · 𝐵) = 0)
42		oveq2 7417	. . . . . 6 ⊢ (𝐵 = 0 → (𝐴 · 𝐵) = (𝐴 · 0))
43		recn 11200	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
44	43	mul01d 11413	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0)
45	44	adantr 482	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 0) = 0)
46	42, 45	sylan9eqr 2795	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐵 = 0) → (𝐴 · 𝐵) = 0)
47	41, 46	jaodan 957	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 = 0 ∨ 𝐵 = 0)) → (𝐴 · 𝐵) = 0)
48	47	ifeq1da 4560	. . 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 2787	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, (𝐴 · 𝐵)))
50	32, 36, 49	3eqtr4d 2783	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ifcif 4529 class class class wbr 5149 (class class class)co 7409 ℝcr 11109 0cc0 11110 · cmul 11115 +∞cpnf 11245 -∞cmnf 11246 ℝ^*cxr 11247 < clt 11248 ·_e cxmu 13091

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-xmul 13094

This theorem is referenced by: xmulrid 13258 xmulgt0 13262 xmulasslem3 13265 xlemul1a 13267 xlemul1 13269 xadddilem 13273 nmoix 24246 nmoi2 24247 metnrmlem3 24377 nmoleub2lem 24630 nn0xmulclb 32015 hashxpe 32050 xrecex 32117 rexdiv 32123 pnfinf 32360 xrge0slmod 32494 esumcst 33092 omssubadd 33330

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