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Theorem rexmul 12658

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 10683	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
2	1	adantr 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≠ +∞)
3	2	necon2bi 3046	. . . . . . . . 9 ⊢ (𝐴 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
4	3	adantl 484	. . . . . . . 8 ⊢ ((0 < 𝐵 ∧ 𝐴 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
5		renemnf 10684	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
6	5	adantr 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≠ -∞)
7	6	necon2bi 3046	. . . . . . . . 9 ⊢ (𝐴 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
8	7	adantl 484	. . . . . . . 8 ⊢ ((𝐵 < 0 ∧ 𝐴 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
9	4, 8	jaoi 853	. . . . . . 7 ⊢ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
10		renepnf 10683	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
11	10	adantl 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≠ +∞)
12	11	necon2bi 3046	. . . . . . . . 9 ⊢ (𝐵 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
13	12	adantl 484	. . . . . . . 8 ⊢ ((0 < 𝐴 ∧ 𝐵 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
14		renemnf 10684	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
15	14	adantl 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≠ -∞)
16	15	necon2bi 3046	. . . . . . . . 9 ⊢ (𝐵 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
17	16	adantl 484	. . . . . . . 8 ⊢ ((𝐴 < 0 ∧ 𝐵 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
18	13, 17	jaoi 853	. . . . . . 7 ⊢ (((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
19	9, 18	jaoi 853	. . . . . 6 ⊢ ((((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
20	19	con2i 141	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (((0 < 𝐵 ∧ 𝐴 = +∞) ∨ (𝐵 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = +∞) ∨ (𝐴 < 0 ∧ 𝐵 = -∞))))
21	20	iffalsed 4477	. . . 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 484	. . . . . . . 8 ⊢ ((0 < 𝐵 ∧ 𝐴 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
23	3	adantl 484	. . . . . . . 8 ⊢ ((𝐵 < 0 ∧ 𝐴 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
24	22, 23	jaoi 853	. . . . . . 7 ⊢ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
25	16	adantl 484	. . . . . . . 8 ⊢ ((0 < 𝐴 ∧ 𝐵 = -∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
26	12	adantl 484	. . . . . . . 8 ⊢ ((𝐴 < 0 ∧ 𝐵 = +∞) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
27	25, 26	jaoi 853	. . . . . . 7 ⊢ (((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞)) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
28	24, 27	jaoi 853	. . . . . 6 ⊢ ((((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
29	28	con2i 141	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ (((0 < 𝐵 ∧ 𝐴 = -∞) ∨ (𝐵 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐵 = -∞) ∨ (𝐴 < 0 ∧ 𝐵 = +∞))))
30	29	iffalsed 4477	. . . 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 2856	. . 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 4485	. 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 10681	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
34		rexr 10681	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
35		xmulval 12612	. . 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 4505	. . 3 ⊢ if((𝐴 = 0 ∨ 𝐵 = 0), (𝐴 · 𝐵), (𝐴 · 𝐵)) = (𝐴 · 𝐵)
38		oveq1 7157	. . . . . 6 ⊢ (𝐴 = 0 → (𝐴 · 𝐵) = (0 · 𝐵))
39		mul02lem2 10811	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (0 · 𝐵) = 0)
40	39	adantl 484	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 · 𝐵) = 0)
41	38, 40	sylan9eqr 2878	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = 0) → (𝐴 · 𝐵) = 0)
42		oveq2 7158	. . . . . 6 ⊢ (𝐵 = 0 → (𝐴 · 𝐵) = (𝐴 · 0))
43		recn 10621	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
44	43	mul01d 10833	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0)
45	44	adantr 483	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 0) = 0)
46	42, 45	sylan9eqr 2878	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐵 = 0) → (𝐴 · 𝐵) = 0)
47	41, 46	jaodan 954	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 = 0 ∨ 𝐵 = 0)) → (𝐴 · 𝐵) = 0)
48	47	ifeq1da 4496	. . 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	syl5eqr 2870	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) = if((𝐴 = 0 ∨ 𝐵 = 0), 0, (𝐴 · 𝐵)))
50	32, 36, 49	3eqtr4d 2866	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ·_e 𝐵) = (𝐴 · 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ifcif 4466 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 0cc0 10531 · cmul 10536 +∞cpnf 10666 -∞cmnf 10667 ℝ^*cxr 10668 < clt 10669 ·_e cxmu 12500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-xmul 12503

This theorem is referenced by: xmulid1 12666 xmulgt0 12670 xmulasslem3 12673 xlemul1a 12675 xlemul1 12677 xadddilem 12681 nmoix 23332 nmoi2 23333 metnrmlem3 23463 nmoleub2lem 23712 nn0xmulclb 30490 hashxpe 30523 xrecex 30591 rexdiv 30597 pnfinf 30807 xrge0slmod 30912 esumcst 31317 omssubadd 31553

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