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Theorem xpncan 13164

Description: Extended real version of pncan 11384. (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 13124	. . . 4 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
2	1	adantl 481	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → -_𝑒𝐵 = -𝐵)
3	2	oveq2d 7372	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐵) = ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵))
4		renegcl 11442	. . . . . 6 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
5	4	ad2antlr 727	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → -𝐵 ∈ ℝ)
6		rexr 11176	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ∈ ℝ^*)
7		renepnf 11178	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ≠ +∞)
8		xaddmnf2 13142	. . . . . 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 7363	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
12		rexr 11176	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
13		renepnf 11178	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
14		xaddmnf2 13142	. . . . . . . 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 2791	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
18	17	oveq1d 7371	. . . 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 2779	. . 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 11179	. . . . . 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 11179	. . . . . 6 ⊢ (-𝐵 ∈ ℝ → -𝐵 ≠ -∞)
29	26, 28	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → -𝐵 ≠ -∞)
30		xaddass 13162	. . . . 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 13147	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 +_𝑒 -𝐵) = (𝐵 + -𝐵))
34	32	recnd 11158	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → 𝐵 ∈ ℂ)
35	34	negidd 11480	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 + -𝐵) = 0)
36	33, 35	eqtrd 2769	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐵 +_𝑒 -𝐵) = 0)
37	36	oveq2d 7372	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)) = (𝐴 +_𝑒 0))
38		xaddrid 13154	. . . . . 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 2769	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 (𝐵 +_𝑒 -𝐵)) = 𝐴)
41	31, 40	eqtrd 2769	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≠ -∞) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
42	20, 41	pm2.61dane 3017	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -𝐵) = 𝐴)
43	3, 42	eqtrd 2769	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 +_𝑒 𝐵) +_𝑒 -_𝑒𝐵) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 (class class class)co 7356 ℝcr 11023 0cc0 11024 + caddc 11027 +∞cpnf 11161 -∞cmnf 11162 ℝ^*cxr 11163 -cneg 11363 -_𝑒cxne 13021 +_𝑒 cxad 13022

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-po 5530 df-so 5531 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-sub 11364 df-neg 11365 df-xneg 13024 df-xadd 13025

This theorem is referenced by: xnpcan 13165 xleadd1 13168 xaddeq0 32782

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