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Theorem xpncan 13284

Description: Extended real version of pncan 11516. (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 13244	. . . 4 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
2	1	adantl 480	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → -_𝑒𝐵 = -𝐵)
3	2	oveq2d 7440	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐵) = ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵))
4		renegcl 11573	. . . . . 6 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
5	4	ad2antlr 725	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → -𝐵 ∈ ℝ)
6		rexr 11310	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ∈ ℝ^*)
7		renepnf 11312	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ≠ +∞)
8		xaddmnf2 13262	. . . . . 6 ⊢ ((-𝐵 ∈ ℝ^* ∧ -𝐵 ≠ +∞) → (-∞ +_𝑒 -𝐵) = -∞)
9	6, 7, 8	syl2anc 582	. . . . 5 ⊢ (-𝐵 ∈ ℝ → (-∞ +_𝑒 -𝐵) = -∞)
10	5, 9	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → (-∞ +_𝑒 -𝐵) = -∞)
11		oveq1 7431	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
12		rexr 11310	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
13		renepnf 11312	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
14		xaddmnf2 13262	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
15	12, 13, 14	syl2anc 582	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (-∞ +_𝑒 𝐵) = -∞)
16	15	adantl 480	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (-∞ +_𝑒 𝐵) = -∞)
17	11, 16	sylan9eqr 2788	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
18	17	oveq1d 7439	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = (-∞ +_𝑒 -𝐵))
19		simpr 483	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → 𝐴 = -∞)
20	10, 18, 19	3eqtr4d 2776	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
21		simpll 765	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐴 ∈ ℝ^*)
22		simpr 483	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐴 ≠ -∞)
23	12	ad2antlr 725	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℝ^*)
24		renemnf 11313	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
25	24	ad2antlr 725	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ≠ -∞)
26	4	ad2antlr 725	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → -𝐵 ∈ ℝ)
27	26, 6	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → -𝐵 ∈ ℝ^*)
28		renemnf 11313	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ≠ -∞)
29	26, 28	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → -𝐵 ≠ -∞)
30		xaddass 13282	. . . . 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 1383	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)))
32		simplr 767	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℝ)
33	32, 26	rexaddd 13267	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 +_𝑒 -𝐵) = (𝐵 + -𝐵))
34	32	recnd 11292	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℂ)
35	34	negidd 11611	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 + -𝐵) = 0)
36	33, 35	eqtrd 2766	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 +_𝑒 -𝐵) = 0)
37	36	oveq2d 7440	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)) = (𝐴 +_𝑒 0))
38		xaddrid 13274	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)
39	38	ad2antrr 724	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 0) = 𝐴)
40	37, 39	eqtrd 2766	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)) = 𝐴)
41	31, 40	eqtrd 2766	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
42	20, 41	pm2.61dane 3019	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
43	3, 42	eqtrd 2766	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐵) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 (class class class)co 7424 ℝcr 11157 0cc0 11158 + caddc 11161 +∞cpnf 11295 -∞cmnf 11296 ℝ^*cxr 11297 -cneg 11495 -_𝑒cxne 13143 +_𝑒 cxad 13144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-sub 11496 df-neg 11497 df-xneg 13146 df-xadd 13147

This theorem is referenced by: xnpcan 13285 xleadd1 13288 xaddeq0 32657

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