Theorem infxrnegsupex 12007

Description: The infimum of a set of extended reals 𝐴 is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)

Hypotheses

Ref	Expression
infxrnegsupex.ex	⊢ (𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
infxrnegsupex.ss	⊢ (𝜑 → 𝐴 ⊆ ℝ*)

Assertion

Ref	Expression
infxrnegsupex	⊢ (𝜑 → inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem infxrnegsupex

Dummy variables 𝑓 𝑔 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrlttri3 10178	. . . . 5 ⊢ ((𝑓 ∈ ℝ* ∧ 𝑔 ∈ ℝ*) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
2	1	adantl 277	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ* ∧ 𝑔 ∈ ℝ*)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓)))
3		infxrnegsupex.ex	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))
4	2, 3	infclti 7353	. . 3 ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ∈ ℝ*)
5		xnegneg 10214	. . 3 ⊢ (inf(𝐴, ℝ*, < ) ∈ ℝ* → -𝑒-𝑒inf(𝐴, ℝ*, < ) = inf(𝐴, ℝ*, < ))
6	4, 5	syl 14	. 2 ⊢ (𝜑 → -𝑒-𝑒inf(𝐴, ℝ*, < ) = inf(𝐴, ℝ*, < ))
7		xnegeq 10208	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → -𝑒𝑤 = -𝑒𝑧)
8	7	cbvmptv 4222	. . . . . . . 8 ⊢ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑧 ∈ ℝ* ↦ -𝑒𝑧)
9	8	mptpreima 5276	. . . . . . 7 ⊢ (◡(𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴) = {𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}
10		eqid 2238	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤)
11	10	xrnegiso 12006	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , ◡ < (ℝ*, ℝ*) ∧ ◡(𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤))
12	11	simpri 113	. . . . . . . 8 ⊢ ◡(𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤)
13	12	imaeq1i 5118	. . . . . . 7 ⊢ (◡(𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴)
14	9, 13	eqtr3i 2261	. . . . . 6 ⊢ {𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴} = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴)
15	14	supeq1i 7318	. . . . 5 ⊢ sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ) = sup(((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴), ℝ*, < )
16	11	simpli 111	. . . . . . . . 9 ⊢ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , ◡ < (ℝ*, ℝ*)
17		isocnv 6007	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom < , ◡ < (ℝ*, ℝ*) → ◡(𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom ◡ < , < (ℝ*, ℝ*))
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ ◡(𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom ◡ < , < (ℝ*, ℝ*)
19		isoeq1 5997	. . . . . . . . 9 ⊢ (◡(𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤) → (◡(𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom ◡ < , < (ℝ*, ℝ*) ↔ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom ◡ < , < (ℝ*, ℝ*)))
20	12, 19	ax-mp 5	. . . . . . . 8 ⊢ (◡(𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom ◡ < , < (ℝ*, ℝ*) ↔ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom ◡ < , < (ℝ*, ℝ*))
21	18, 20	mpbi 145	. . . . . . 7 ⊢ (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom ◡ < , < (ℝ*, ℝ*)
22	21	a1i 9	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ℝ* ↦ -𝑒𝑤) Isom ◡ < , < (ℝ*, ℝ*))
23		infxrnegsupex.ss	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ*)
24	3	cnvinfex 7348	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥◡ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦◡ < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦◡ < 𝑧)))
25	2	cnvti 7349	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ ℝ* ∧ 𝑔 ∈ ℝ*)) → (𝑓 = 𝑔 ↔ (¬ 𝑓◡ < 𝑔 ∧ ¬ 𝑔◡ < 𝑓)))
26	22, 23, 24, 25	supisoti 7340	. . . . 5 ⊢ (𝜑 → sup(((𝑤 ∈ ℝ* ↦ -𝑒𝑤) “ 𝐴), ℝ*, < ) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, ◡ < )))
27	15, 26	eqtrid 2283	. . . 4 ⊢ (𝜑 → sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, ◡ < )))
28		df-inf 7315	. . . . . . 7 ⊢ inf(𝐴, ℝ*, < ) = sup(𝐴, ℝ*, ◡ < )
29	28	eqcomi 2242	. . . . . 6 ⊢ sup(𝐴, ℝ*, ◡ < ) = inf(𝐴, ℝ*, < )
30	29	fveq2i 5693	. . . . 5 ⊢ ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, ◡ < )) = ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘inf(𝐴, ℝ*, < ))
31		eqidd 2239	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ ℝ* ↦ -𝑒𝑤) = (𝑤 ∈ ℝ* ↦ -𝑒𝑤))
32		xnegeq 10208	. . . . . . 7 ⊢ (𝑤 = inf(𝐴, ℝ*, < ) → -𝑒𝑤 = -𝑒inf(𝐴, ℝ*, < ))
33	32	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = inf(𝐴, ℝ*, < )) → -𝑒𝑤 = -𝑒inf(𝐴, ℝ*, < ))
34	4	xnegcld 10236	. . . . . 6 ⊢ (𝜑 → -𝑒inf(𝐴, ℝ*, < ) ∈ ℝ*)
35	31, 33, 4, 34	fvmptd 5780	. . . . 5 ⊢ (𝜑 → ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘inf(𝐴, ℝ*, < )) = -𝑒inf(𝐴, ℝ*, < ))
36	30, 35	eqtrid 2283	. . . 4 ⊢ (𝜑 → ((𝑤 ∈ ℝ* ↦ -𝑒𝑤)‘sup(𝐴, ℝ*, ◡ < )) = -𝑒inf(𝐴, ℝ*, < ))
37	27, 36	eqtr2d 2272	. . 3 ⊢ (𝜑 → -𝑒inf(𝐴, ℝ*, < ) = sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ))
38		xnegeq 10208	. . 3 ⊢ (-𝑒inf(𝐴, ℝ*, < ) = sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ) → -𝑒-𝑒inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ))
39	37, 38	syl 14	. 2 ⊢ (𝜑 → -𝑒-𝑒inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ))
40	6, 39	eqtr3d 2273	1 ⊢ (𝜑 → inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532   ⊆ wss 3220   class class class wbr 4125   ↦ cmpt 4187  ^◡ccnv 4768   “ cima 4772  ‘cfv 5372   Isom wiso 5373  supcsup 7312  infcinf 7313  ℝ^*cxr 8349   < clt 8350  -_𝑒cxne 10150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-sub 8489  df-neg 8490  df-xneg 10153

This theorem is referenced by:  xrminmax  12009