xrge0npcan - Mathbox for Thierry Arnoux

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Theorem xrge0npcan 30676

Description: Extended nonnegative real version of npcan 10889. (Contributed by Thierry Arnoux, 9-Jun-2017.)

**Assertion**
Ref	Expression
xrge0npcan	⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐴)

**Proof of Theorem xrge0npcan**
Step	Hyp	Ref	Expression
1		iccssxr 12813	. . . . . . . . 9 ⊢ (0[,]+∞) ⊆ ℝ^*
2		simpl1 1187	. . . . . . . . 9 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → 𝐴 ∈ (0[,]+∞))
3	1, 2	sseldi 3965	. . . . . . . 8 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → 𝐴 ∈ ℝ^*)
4		simpr 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → 𝐵 = +∞)
5		simpl3 1189	. . . . . . . . 9 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → 𝐵 ≤ 𝐴)
6	4, 5	eqbrtrrd 5083	. . . . . . . 8 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → +∞ ≤ 𝐴)
7		xgepnf 12552	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ^* → (+∞ ≤ 𝐴 ↔ 𝐴 = +∞))
8	7	biimpa 479	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ≤ 𝐴) → 𝐴 = +∞)
9	3, 6, 8	syl2anc 586	. . . . . . 7 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → 𝐴 = +∞)
10		xnegeq 12594	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
11	4, 10	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → -_𝑒𝐵 = -_𝑒+∞)
12	9, 11	oveq12d 7168	. . . . . 6 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (+∞ +_𝑒 -_𝑒+∞))
13		pnfxr 10689	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
14		xnegid 12625	. . . . . . 7 ⊢ (+∞ ∈ ℝ^* → (+∞ +_𝑒 -_𝑒+∞) = 0)
15	13, 14	ax-mp 5	. . . . . 6 ⊢ (+∞ +_𝑒 -_𝑒+∞) = 0
16	12, 15	syl6eq 2872	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 -_𝑒𝐵) = 0)
17	16	oveq1d 7165	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = (0 +_𝑒 𝐵))
18	4	oveq2d 7166	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → (0 +_𝑒 𝐵) = (0 +_𝑒 +∞))
19		xaddid2 12629	. . . . 5 ⊢ (+∞ ∈ ℝ^* → (0 +_𝑒 +∞) = +∞)
20	13, 19	mp1i 13	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → (0 +_𝑒 +∞) = +∞)
21	17, 18, 20	3eqtrd 2860	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = +∞)
22	21, 9	eqtr4d 2859	. 2 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐴)
23		simpl1 1187	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐴 ∈ (0[,]+∞))
24	1, 23	sseldi 3965	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐴 ∈ ℝ^*)
25		xrge0neqmnf 12834	. . . . 5 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞)
26	23, 25	syl 17	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐴 ≠ -∞)
27		simpl2 1188	. . . . . 6 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ (0[,]+∞))
28	1, 27	sseldi 3965	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ ℝ^*)
29	28	xnegcld 12687	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → -_𝑒𝐵 ∈ ℝ^*)
30		simpr 487	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → ¬ 𝐵 = +∞)
31		xnegneg 12601	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
32		xnegeq 12594	. . . . . . . . 9 ⊢ (-_𝑒𝐵 = -∞ → -_𝑒-_𝑒𝐵 = -_𝑒-∞)
33	31, 32	sylan9req 2877	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 = -∞) → 𝐵 = -_𝑒-∞)
34		xnegmnf 12597	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
35	33, 34	syl6eq 2872	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 = -∞) → 𝐵 = +∞)
36	35	stoic1a 1769	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ 𝐵 = +∞) → ¬ -_𝑒𝐵 = -∞)
37	36	neqned 3023	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ 𝐵 = +∞) → -_𝑒𝐵 ≠ -∞)
38	28, 30, 37	syl2anc 586	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → -_𝑒𝐵 ≠ -∞)
39		xrge0neqmnf 12834	. . . . 5 ⊢ (𝐵 ∈ (0[,]+∞) → 𝐵 ≠ -∞)
40	27, 39	syl 17	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ -∞)
41		xaddass 12636	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) ∧ (-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 ≠ -∞) ∧ (𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞)) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = (𝐴 +_𝑒 (-_𝑒𝐵 +_𝑒 𝐵)))
42	24, 26, 29, 38, 28, 40, 41	syl222anc 1382	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = (𝐴 +_𝑒 (-_𝑒𝐵 +_𝑒 𝐵)))
43		xnegcl 12600	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
44		xaddcom 12627	. . . . . . . 8 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐵 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐵))
45	43, 44	mpancom 686	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐵))
46		xnegid 12625	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → (𝐵 +_𝑒 -_𝑒𝐵) = 0)
47	45, 46	eqtrd 2856	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 +_𝑒 𝐵) = 0)
48	47	oveq2d 7166	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐴 +_𝑒 (-_𝑒𝐵 +_𝑒 𝐵)) = (𝐴 +_𝑒 0))
49		xaddid1 12628	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)
50	48, 49	sylan9eqr 2878	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 (-_𝑒𝐵 +_𝑒 𝐵)) = 𝐴)
51	24, 28, 50	syl2anc 586	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → (𝐴 +_𝑒 (-_𝑒𝐵 +_𝑒 𝐵)) = 𝐴)
52	42, 51	eqtrd 2856	. 2 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐴)
53	22, 52	pm2.61dan 811	1 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5059 (class class class)co 7150 0cc0 10531 +∞cpnf 10666 -∞cmnf 10667 ℝ^*cxr 10668 ≤ cle 10670 -_𝑒cxne 12498 +_𝑒 cxad 12499 [,]cicc 12735

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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 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-le 10675 df-sub 10866 df-neg 10867 df-xneg 12501 df-xadd 12502 df-icc 12739

This theorem is referenced by: esumle 31312 esumlef 31316 carsgclctunlem2 31572

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