xrge0npcan - Mathbox for Thierry Arnoux

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Theorem xrge0npcan 30685

Description: Extended nonnegative real version of npcan 10898. (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 12822	. . . . . . . . 9 ⊢ (0[,]+∞) ⊆ ℝ^*
2		simpl1 1187	. . . . . . . . 9 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → 𝐴 ∈ (0[,]+∞))
3	1, 2	sseldi 3968	. . . . . . . 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 5093	. . . . . . . 8 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → +∞ ≤ 𝐴)
7		xgepnf 12561	. . . . . . . . 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 12603	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
11	4, 10	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → -_𝑒𝐵 = -_𝑒+∞)
12	9, 11	oveq12d 7177	. . . . . 6 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 -_𝑒𝐵) = (+∞ +_𝑒 -_𝑒+∞))
13		pnfxr 10698	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
14		xnegid 12634	. . . . . . 7 ⊢ (+∞ ∈ ℝ^* → (+∞ +_𝑒 -_𝑒+∞) = 0)
15	13, 14	ax-mp 5	. . . . . 6 ⊢ (+∞ +_𝑒 -_𝑒+∞) = 0
16	12, 15	syl6eq 2875	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → (𝐴 +_𝑒 -_𝑒𝐵) = 0)
17	16	oveq1d 7174	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = (0 +_𝑒 𝐵))
18	4	oveq2d 7175	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → (0 +_𝑒 𝐵) = (0 +_𝑒 +∞))
19		xaddid2 12638	. . . . 5 ⊢ (+∞ ∈ ℝ^* → (0 +_𝑒 +∞) = +∞)
20	13, 19	mp1i 13	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → (0 +_𝑒 +∞) = +∞)
21	17, 18, 20	3eqtrd 2863	. . 3 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ 𝐵 = +∞) → ((𝐴 +_𝑒 -_𝑒𝐵) +_𝑒 𝐵) = +∞)
22	21, 9	eqtr4d 2862	. 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 3968	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐴 ∈ ℝ^*)
25		xrge0neqmnf 12843	. . . . 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 3968	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ ℝ^*)
29	28	xnegcld 12696	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → -_𝑒𝐵 ∈ ℝ^*)
30		simpr 487	. . . . 5 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → ¬ 𝐵 = +∞)
31		xnegneg 12610	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
32		xnegeq 12603	. . . . . . . . 9 ⊢ (-_𝑒𝐵 = -∞ → -_𝑒-_𝑒𝐵 = -_𝑒-∞)
33	31, 32	sylan9req 2880	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 = -∞) → 𝐵 = -_𝑒-∞)
34		xnegmnf 12606	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
35	33, 34	syl6eq 2875	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 = -∞) → 𝐵 = +∞)
36	35	stoic1a 1772	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ 𝐵 = +∞) → ¬ -_𝑒𝐵 = -∞)
37	36	neqned 3026	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ ¬ 𝐵 = +∞) → -_𝑒𝐵 ≠ -∞)
38	28, 30, 37	syl2anc 586	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → -_𝑒𝐵 ≠ -∞)
39		xrge0neqmnf 12843	. . . . 5 ⊢ (𝐵 ∈ (0[,]+∞) → 𝐵 ≠ -∞)
40	27, 39	syl 17	. . . 4 ⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ≠ -∞)
41		xaddass 12645	. . . 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 12609	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
44		xaddcom 12636	. . . . . . . 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 12634	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → (𝐵 +_𝑒 -_𝑒𝐵) = 0)
47	45, 46	eqtrd 2859	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 +_𝑒 𝐵) = 0)
48	47	oveq2d 7175	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐴 +_𝑒 (-_𝑒𝐵 +_𝑒 𝐵)) = (𝐴 +_𝑒 0))
49		xaddid1 12637	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)
50	48, 49	sylan9eqr 2881	. . . 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 2859	. 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 1536 ∈ wcel 2113 ≠ wne 3019 class class class wbr 5069 (class class class)co 7159 0cc0 10540 +∞cpnf 10675 -∞cmnf 10676 ℝ^*cxr 10677 ≤ cle 10679 -_𝑒cxne 12507 +_𝑒 cxad 12508 [,]cicc 12744

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-xneg 12510 df-xadd 12511 df-icc 12748

This theorem is referenced by: esumle 31321 esumlef 31325 carsgclctunlem2 31581

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