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Theorem xpncan 13256

Description: Extended real version of pncan 11438. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xpncan	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐵) = 𝐴)

**Proof of Theorem xpncan**
Step	Hyp	Ref	Expression
1		rexneg 13216	. . . 4 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
2	1	adantl 485	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → -_𝑒𝐵 = -𝐵)
3	2	oveq2d 7414	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐵) = ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵))
4		renegcl 11496	. . . . . 6 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
5	4	ad2antlr 737	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → -𝐵 ∈ ℝ)
6		rexr 11230	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ∈ ℝ^*)
7		renepnf 11232	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ≠ +∞)
8		xaddmnf2 13234	. . . . . 6 ⊢ ((-𝐵 ∈ ℝ^* ∧ -𝐵 ≠ +∞) → (-∞ +_𝑒 -𝐵) = -∞)
9	6, 7, 8	syl2anc 593	. . . . 5 ⊢ (-𝐵 ∈ ℝ → (-∞ +_𝑒 -𝐵) = -∞)
10	5, 9	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → (-∞ +_𝑒 -𝐵) = -∞)
11		oveq1 7405	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
12		rexr 11230	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
13		renepnf 11232	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
14		xaddmnf2 13234	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
15	12, 13, 14	syl2anc 593	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (-∞ +_𝑒 𝐵) = -∞)
16	15	adantl 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (-∞ +_𝑒 𝐵) = -∞)
17	11, 16	sylan9eqr 2821	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
18	17	oveq1d 7413	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = (-∞ +_𝑒 -𝐵))
19		simpr 488	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → 𝐴 = -∞)
20	10, 18, 19	3eqtr4d 2809	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
21		simpll 776	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐴 ∈ ℝ^*)
22		simpr 488	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐴 ≠ -∞)
23	12	ad2antlr 737	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℝ^*)
24		renemnf 11233	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
25	24	ad2antlr 737	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ≠ -∞)
26	4	ad2antlr 737	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → -𝐵 ∈ ℝ)
27	26, 6	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → -𝐵 ∈ ℝ^*)
28		renemnf 11233	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ≠ -∞)
29	26, 28	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → -𝐵 ≠ -∞)
30		xaddass 13254	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) ∧ (𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) ∧ (-𝐵 ∈ ℝ^* ∧ -𝐵 ≠ -∞)) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)))
31	21, 22, 23, 25, 27, 29, 30	syl222anc 1407	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)))
32		simplr 778	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℝ)
33	32, 26	rexaddd 13239	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 +_𝑒 -𝐵) = (𝐵 + -𝐵))
34	32	recnd 11212	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℂ)
35	34	negidd 11534	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 + -𝐵) = 0)
36	33, 35	eqtrd 2799	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 +_𝑒 -𝐵) = 0)
37	36	oveq2d 7414	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)) = (𝐴 +_𝑒 0))
38		xaddrid 13246	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)
39	38	ad2antrr 736	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 0) = 𝐴)
40	37, 39	eqtrd 2799	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)) = 𝐴)
41	31, 40	eqtrd 2799	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
42	20, 41	pm2.61dane 3046	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
43	3, 42	eqtrd 2799	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐵) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1562 ∈ wcel 2144 ≠ wne 2959 (class class class)co 7398 ℝcr 11074 0cc0 11075 + caddc 11078 +∞cpnf 11215 -∞cmnf 11216 ℝ^*cxr 11217 -cneg 11417 -_𝑒cxne 13113 +_𝑒 cxad 13114

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5544 df-po 5557 df-so 5558 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-1st 7972 df-2nd 7973 df-er 8680 df-en 8930 df-dom 8931 df-sdom 8932 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-sub 11418 df-neg 11419 df-xneg 13116 df-xadd 13117

This theorem is referenced by: xnpcan 13257 xleadd1 13260 xaddeq0 32957

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