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Theorem xpncan 13211

Description: Extended real version of pncan 11427. (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 13171	. . . 4 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
2	1	adantl 481	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → -_𝑒𝐵 = -𝐵)
3	2	oveq2d 7403	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐵) = ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵))
4		renegcl 11485	. . . . . 6 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
5	4	ad2antlr 727	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → -𝐵 ∈ ℝ)
6		rexr 11220	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ∈ ℝ^*)
7		renepnf 11222	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ≠ +∞)
8		xaddmnf2 13189	. . . . . 6 ⊢ ((-𝐵 ∈ ℝ^* ∧ -𝐵 ≠ +∞) → (-∞ +_𝑒 -𝐵) = -∞)
9	6, 7, 8	syl2anc 584	. . . . 5 ⊢ (-𝐵 ∈ ℝ → (-∞ +_𝑒 -𝐵) = -∞)
10	5, 9	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → (-∞ +_𝑒 -𝐵) = -∞)
11		oveq1 7394	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
12		rexr 11220	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
13		renepnf 11222	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
14		xaddmnf2 13189	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
15	12, 13, 14	syl2anc 584	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (-∞ +_𝑒 𝐵) = -∞)
16	15	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (-∞ +_𝑒 𝐵) = -∞)
17	11, 16	sylan9eqr 2786	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
18	17	oveq1d 7402	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = (-∞ +_𝑒 -𝐵))
19		simpr 484	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → 𝐴 = -∞)
20	10, 18, 19	3eqtr4d 2774	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
21		simpll 766	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐴 ∈ ℝ^*)
22		simpr 484	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐴 ≠ -∞)
23	12	ad2antlr 727	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℝ^*)
24		renemnf 11223	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
25	24	ad2antlr 727	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ≠ -∞)
26	4	ad2antlr 727	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → -𝐵 ∈ ℝ)
27	26, 6	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → -𝐵 ∈ ℝ^*)
28		renemnf 11223	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ≠ -∞)
29	26, 28	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → -𝐵 ≠ -∞)
30		xaddass 13209	. . . . 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 1388	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)))
32		simplr 768	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℝ)
33	32, 26	rexaddd 13194	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 +_𝑒 -𝐵) = (𝐵 + -𝐵))
34	32	recnd 11202	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℂ)
35	34	negidd 11523	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 + -𝐵) = 0)
36	33, 35	eqtrd 2764	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 +_𝑒 -𝐵) = 0)
37	36	oveq2d 7403	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)) = (𝐴 +_𝑒 0))
38		xaddrid 13201	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)
39	38	ad2antrr 726	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 0) = 𝐴)
40	37, 39	eqtrd 2764	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)) = 𝐴)
41	31, 40	eqtrd 2764	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
42	20, 41	pm2.61dane 3012	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
43	3, 42	eqtrd 2764	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐵) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 (class class class)co 7387 ℝcr 11067 0cc0 11068 + caddc 11071 +∞cpnf 11205 -∞cmnf 11206 ℝ^*cxr 11207 -cneg 11406 -_𝑒cxne 13069 +_𝑒 cxad 13070

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-reu 3355 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-iun 4957 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-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-1st 7968 df-2nd 7969 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-sub 11407 df-neg 11408 df-xneg 13072 df-xadd 13073

This theorem is referenced by: xnpcan 13212 xleadd1 13215 xaddeq0 32676

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