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Theorem supminfrnmpt 39985
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 2651 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supminfrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 39699 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 supminfrnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 39727 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 supminfrnmpt.y . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
81, 3rnmptbd 39785 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
97, 8mpbid 222 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
10 supminf 11813 . . 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 1366 . 2 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
12 eqid 2651 . . . . . . . . 9 (𝑥𝐴 ↦ -𝐵) = (𝑥𝐴 ↦ -𝐵)
13 simpr 476 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → -𝑤 ∈ ran (𝑥𝐴𝐵))
14 renegcl 10382 . . . . . . . . . . . . . 14 (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
152elrnmpt 5404 . . . . . . . . . . . . . 14 (-𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1614, 15syl 17 . . . . . . . . . . . . 13 (𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1716adantr 480 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1813, 17mpbid 222 . . . . . . . . . . 11 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 -𝑤 = 𝐵)
1918adantll 750 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 -𝑤 = 𝐵)
20 nfv 1883 . . . . . . . . . . . . 13 𝑥 𝑤 ∈ ℝ
211, 20nfan 1868 . . . . . . . . . . . 12 𝑥(𝜑𝑤 ∈ ℝ)
22 negeq 10311 . . . . . . . . . . . . . . . . . . 19 (-𝑤 = 𝐵 → --𝑤 = -𝐵)
2322eqcomd 2657 . . . . . . . . . . . . . . . . . 18 (-𝑤 = 𝐵 → -𝐵 = --𝑤)
2423adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
25 recn 10064 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
2625negnegd 10421 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ℝ → --𝑤 = 𝑤)
2726adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
2824, 27eqtr2d 2686 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
2928ex 449 . . . . . . . . . . . . . . 15 (𝑤 ∈ ℝ → (-𝑤 = 𝐵𝑤 = -𝐵))
3029adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ℝ) → (-𝑤 = 𝐵𝑤 = -𝐵))
3130adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (-𝑤 = 𝐵𝑤 = -𝐵))
32 negeq 10311 . . . . . . . . . . . . . . . . 17 (𝑤 = -𝐵 → -𝑤 = --𝐵)
3332adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
343recnd 10106 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
3534negnegd 10421 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → --𝐵 = 𝐵)
3635adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
3733, 36eqtrd 2685 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
3837ex 449 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
3938adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
4031, 39impbid 202 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (-𝑤 = 𝐵𝑤 = -𝐵))
4121, 40rexbida 3076 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ℝ) → (∃𝑥𝐴 -𝑤 = 𝐵 ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4241adantr 480 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 -𝑤 = 𝐵 ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4319, 42mpbid 222 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑤 = -𝐵)
44 simplr 807 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → 𝑤 ∈ ℝ)
4512, 43, 44elrnmptd 39680 . . . . . . . 8 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵))
4645ex 449 . . . . . . 7 ((𝜑𝑤 ∈ ℝ) → (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
4746ralrimiva 2995 . . . . . 6 (𝜑 → ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
48 rabss 3712 . . . . . 6 ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝐵) ↔ ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
4947, 48sylibr 224 . . . . 5 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝐵))
50 nfcv 2793 . . . . . . . 8 𝑥-𝑤
51 nfmpt1 4780 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
5251nfrn 5400 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
5350, 52nfel 2806 . . . . . . 7 𝑥-𝑤 ∈ ran (𝑥𝐴𝐵)
54 nfcv 2793 . . . . . . 7 𝑥
5553, 54nfrab 3153 . . . . . 6 𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
5632eleq1d 2715 . . . . . . 7 (𝑤 = -𝐵 → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ --𝐵 ∈ ran (𝑥𝐴𝐵)))
573renegcld 10495 . . . . . . 7 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
58 simpr 476 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
592elrnmpt1 5406 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥𝐴𝐵))
6058, 3, 59syl2anc 694 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
6135, 60eqeltrd 2730 . . . . . . 7 ((𝜑𝑥𝐴) → --𝐵 ∈ ran (𝑥𝐴𝐵))
6256, 57, 61elrabd 3398 . . . . . 6 ((𝜑𝑥𝐴) → -𝐵 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
631, 55, 12, 62rnmptssdf 39783 . . . . 5 (𝜑 → ran (𝑥𝐴 ↦ -𝐵) ⊆ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
6449, 63eqssd 3653 . . . 4 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵))
6564infeq1d 8424 . . 3 (𝜑 → inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
6665negeqd 10313 . 2 (𝜑 → -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
6711, 66eqtrd 2685 1 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wnf 1748  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  wss 3607  c0 3948   class class class wbr 4685  cmpt 4762  ran crn 5144  supcsup 8387  infcinf 8388  cr 9973   < clt 10112  cle 10113  -cneg 10305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307
This theorem is referenced by: (None)
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