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Theorem supminfrnmpt 42448
 Description: The indexed supremum of a bounded-above set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
supminfrnmpt.x 𝑥𝜑
supminfrnmpt.a (𝜑𝐴 ≠ ∅)
supminfrnmpt.b ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
supminfrnmpt.y (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
Assertion
Ref Expression
supminfrnmpt (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem supminfrnmpt
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supminfrnmpt.x . . . 4 𝑥𝜑
2 eqid 2758 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supminfrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 42194 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 supminfrnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 6073 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 supminfrnmpt.y . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
81, 3rnmptbd 42262 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
97, 8mpbid 235 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
10 supminf 12375 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ ∧ ran (𝑥𝐴𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
114, 6, 9, 10syl3anc 1368 . 2 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
12 eqid 2758 . . . . . . . . 9 (𝑥𝐴 ↦ -𝐵) = (𝑥𝐴 ↦ -𝐵)
13 simpr 488 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → -𝑤 ∈ ran (𝑥𝐴𝐵))
14 renegcl 10987 . . . . . . . . . . . . . 14 (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
152elrnmpt 5797 . . . . . . . . . . . . . 14 (-𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1614, 15syl 17 . . . . . . . . . . . . 13 (𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1716adantr 484 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1813, 17mpbid 235 . . . . . . . . . . 11 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 -𝑤 = 𝐵)
1918adantll 713 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 -𝑤 = 𝐵)
20 nfv 1915 . . . . . . . . . . . . 13 𝑥 𝑤 ∈ ℝ
211, 20nfan 1900 . . . . . . . . . . . 12 𝑥(𝜑𝑤 ∈ ℝ)
22 negeq 10916 . . . . . . . . . . . . . . . . . . 19 (-𝑤 = 𝐵 → --𝑤 = -𝐵)
2322eqcomd 2764 . . . . . . . . . . . . . . . . . 18 (-𝑤 = 𝐵 → -𝐵 = --𝑤)
2423adantl 485 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
25 recn 10665 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
2625negnegd 11026 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ℝ → --𝑤 = 𝑤)
2726adantr 484 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
2824, 27eqtr2d 2794 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
2928ex 416 . . . . . . . . . . . . . . 15 (𝑤 ∈ ℝ → (-𝑤 = 𝐵𝑤 = -𝐵))
3029adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ℝ) → (-𝑤 = 𝐵𝑤 = -𝐵))
3130adantr 484 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (-𝑤 = 𝐵𝑤 = -𝐵))
32 negeq 10916 . . . . . . . . . . . . . . . . 17 (𝑤 = -𝐵 → -𝑤 = --𝐵)
3332adantl 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
343recnd 10707 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
3534negnegd 11026 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → --𝐵 = 𝐵)
3635adantr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
3733, 36eqtrd 2793 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
3837ex 416 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
3938adantlr 714 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
4031, 39impbid 215 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (-𝑤 = 𝐵𝑤 = -𝐵))
4121, 40rexbida 3242 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ℝ) → (∃𝑥𝐴 -𝑤 = 𝐵 ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4241adantr 484 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 -𝑤 = 𝐵 ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4319, 42mpbid 235 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑤 = -𝐵)
44 simplr 768 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → 𝑤 ∈ ℝ)
4512, 43, 44elrnmptd 5802 . . . . . . . 8 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵))
4645ex 416 . . . . . . 7 ((𝜑𝑤 ∈ ℝ) → (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
4746ralrimiva 3113 . . . . . 6 (𝜑 → ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
48 rabss 3976 . . . . . 6 ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝐵) ↔ ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
4947, 48sylibr 237 . . . . 5 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝐵))
50 nfcv 2919 . . . . . . . 8 𝑥-𝑤
51 nfmpt1 5130 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
5251nfrn 5793 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
5350, 52nfel 2933 . . . . . . 7 𝑥-𝑤 ∈ ran (𝑥𝐴𝐵)
54 nfcv 2919 . . . . . . 7 𝑥
5553, 54nfrabw 3303 . . . . . 6 𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
5632eleq1d 2836 . . . . . . 7 (𝑤 = -𝐵 → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ --𝐵 ∈ ran (𝑥𝐴𝐵)))
573renegcld 11105 . . . . . . 7 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
58 simpr 488 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
592elrnmpt1 5799 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥𝐴𝐵))
6058, 3, 59syl2anc 587 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
6135, 60eqeltrd 2852 . . . . . . 7 ((𝜑𝑥𝐴) → --𝐵 ∈ ran (𝑥𝐴𝐵))
6256, 57, 61elrabd 3604 . . . . . 6 ((𝜑𝑥𝐴) → -𝐵 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
631, 55, 12, 62rnmptssdf 42260 . . . . 5 (𝜑 → ran (𝑥𝐴 ↦ -𝐵) ⊆ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
6449, 63eqssd 3909 . . . 4 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵))
6564infeq1d 8974 . . 3 (𝜑 → inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
6665negeqd 10918 . 2 (𝜑 → -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
6711, 66eqtrd 2793 1 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074   ⊆ wss 3858  ∅c0 4225   class class class wbr 5032   ↦ cmpt 5112  ran crn 5525  supcsup 8937  infcinf 8938  ℝcr 10574   < clt 10713   ≤ cle 10714  -cneg 10909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-po 5443  df-so 5444  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-sup 8939  df-inf 8940  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911 This theorem is referenced by: (None)
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