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Theorem supminfrnmpt 40149
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 2804 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supminfrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 39872 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 supminfrnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 39898 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 supminfrnmpt.y . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
81, 3rnmptbd 39952 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦))
97, 8mpbid 223 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
10 supminf 11992 . . 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 1483 . 2 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
12 eqid 2804 . . . . . . . . 9 (𝑥𝐴 ↦ -𝐵) = (𝑥𝐴 ↦ -𝐵)
13 simpr 473 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → -𝑤 ∈ ran (𝑥𝐴𝐵))
14 renegcl 10627 . . . . . . . . . . . . . 14 (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
152elrnmpt 5571 . . . . . . . . . . . . . 14 (-𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1614, 15syl 17 . . . . . . . . . . . . 13 (𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1716adantr 468 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1813, 17mpbid 223 . . . . . . . . . . 11 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 -𝑤 = 𝐵)
1918adantll 696 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 -𝑤 = 𝐵)
20 nfv 2005 . . . . . . . . . . . . 13 𝑥 𝑤 ∈ ℝ
211, 20nfan 1990 . . . . . . . . . . . 12 𝑥(𝜑𝑤 ∈ ℝ)
22 negeq 10556 . . . . . . . . . . . . . . . . . . 19 (-𝑤 = 𝐵 → --𝑤 = -𝐵)
2322eqcomd 2810 . . . . . . . . . . . . . . . . . 18 (-𝑤 = 𝐵 → -𝐵 = --𝑤)
2423adantl 469 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
25 recn 10309 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
2625negnegd 10666 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ℝ → --𝑤 = 𝑤)
2726adantr 468 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
2824, 27eqtr2d 2839 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
2928ex 399 . . . . . . . . . . . . . . 15 (𝑤 ∈ ℝ → (-𝑤 = 𝐵𝑤 = -𝐵))
3029adantl 469 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ℝ) → (-𝑤 = 𝐵𝑤 = -𝐵))
3130adantr 468 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (-𝑤 = 𝐵𝑤 = -𝐵))
32 negeq 10556 . . . . . . . . . . . . . . . . 17 (𝑤 = -𝐵 → -𝑤 = --𝐵)
3332adantl 469 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
343recnd 10351 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
3534negnegd 10666 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → --𝐵 = 𝐵)
3635adantr 468 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
3733, 36eqtrd 2838 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
3837ex 399 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
3938adantlr 697 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
4031, 39impbid 203 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (-𝑤 = 𝐵𝑤 = -𝐵))
4121, 40rexbida 3233 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ℝ) → (∃𝑥𝐴 -𝑤 = 𝐵 ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4241adantr 468 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 -𝑤 = 𝐵 ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4319, 42mpbid 223 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑤 = -𝐵)
44 simplr 776 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → 𝑤 ∈ ℝ)
4512, 43, 44elrnmptd 39853 . . . . . . . 8 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵))
4645ex 399 . . . . . . 7 ((𝜑𝑤 ∈ ℝ) → (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
4746ralrimiva 3152 . . . . . 6 (𝜑 → ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
48 rabss 3874 . . . . . 6 ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝐵) ↔ ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
4947, 48sylibr 225 . . . . 5 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝐵))
50 nfcv 2946 . . . . . . . 8 𝑥-𝑤
51 nfmpt1 4939 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
5251nfrn 5567 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
5350, 52nfel 2959 . . . . . . 7 𝑥-𝑤 ∈ ran (𝑥𝐴𝐵)
54 nfcv 2946 . . . . . . 7 𝑥
5553, 54nfrab 3310 . . . . . 6 𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
5632eleq1d 2868 . . . . . . 7 (𝑤 = -𝐵 → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ --𝐵 ∈ ran (𝑥𝐴𝐵)))
573renegcld 10740 . . . . . . 7 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
58 simpr 473 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
592elrnmpt1 5573 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥𝐴𝐵))
6058, 3, 59syl2anc 575 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
6135, 60eqeltrd 2883 . . . . . . 7 ((𝜑𝑥𝐴) → --𝐵 ∈ ran (𝑥𝐴𝐵))
6256, 57, 61elrabd 3559 . . . . . 6 ((𝜑𝑥𝐴) → -𝐵 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
631, 55, 12, 62rnmptssdf 39950 . . . . 5 (𝜑 → ran (𝑥𝐴 ↦ -𝐵) ⊆ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
6449, 63eqssd 3813 . . . 4 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵))
6564infeq1d 8620 . . 3 (𝜑 → inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
6665negeqd 10558 . 2 (𝜑 → -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
6711, 66eqtrd 2838 1 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wnf 1863  wcel 2156  wne 2976  wral 3094  wrex 3095  {crab 3098  wss 3767  c0 4114   class class class wbr 4842  cmpt 4921  ran crn 5310  supcsup 8583  infcinf 8584  cr 10218   < clt 10357  cle 10358  -cneg 10550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177  ax-resscn 10276  ax-1cn 10277  ax-icn 10278  ax-addcl 10279  ax-addrcl 10280  ax-mulcl 10281  ax-mulrcl 10282  ax-mulcom 10283  ax-addass 10284  ax-mulass 10285  ax-distr 10286  ax-i2m1 10287  ax-1ne0 10288  ax-1rid 10289  ax-rnegex 10290  ax-rrecex 10291  ax-cnre 10292  ax-pre-lttri 10293  ax-pre-lttrn 10294  ax-pre-ltadd 10295  ax-pre-mulgt0 10296  ax-pre-sup 10297
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ne 2977  df-nel 3080  df-ral 3099  df-rex 3100  df-reu 3101  df-rmo 3102  df-rab 3103  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4115  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5217  df-po 5230  df-so 5231  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-rn 5320  df-res 5321  df-ima 5322  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-isom 6108  df-riota 6833  df-ov 6875  df-oprab 6876  df-mpt2 6877  df-er 7977  df-en 8191  df-dom 8192  df-sdom 8193  df-sup 8585  df-inf 8586  df-pnf 10359  df-mnf 10360  df-xr 10361  df-ltxr 10362  df-le 10363  df-sub 10551  df-neg 10552
This theorem is referenced by: (None)
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