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Theorem supminfrnmpt 44886
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 2725 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3 supminfrnmpt.b . . . 4 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
41, 2, 3rnmptssd 44629 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ⊆ ℝ)
5 supminfrnmpt.a . . . 4 (𝜑𝐴 ≠ ∅)
61, 3, 2, 5rnmptn0 6244 . . 3 (𝜑 → ran (𝑥𝐴𝐵) ≠ ∅)
7 supminfrnmpt.y . . . 4 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥𝐴 𝐵𝑦)
81, 7rnmptbdd 44680 . . 3 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦)
9 supminf 12944 . . 3 ((ran (𝑥𝐴𝐵) ⊆ ℝ ∧ ran (𝑥𝐴𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥𝐴𝐵)𝑧𝑦) → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
104, 6, 8, 9syl3anc 1368 . 2 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ))
11 eqid 2725 . . . . . . . . 9 (𝑥𝐴 ↦ -𝐵) = (𝑥𝐴 ↦ -𝐵)
12 simpr 483 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → -𝑤 ∈ ran (𝑥𝐴𝐵))
13 renegcl 11548 . . . . . . . . . . . . . 14 (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
142elrnmpt 5953 . . . . . . . . . . . . . 14 (-𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1513, 14syl 17 . . . . . . . . . . . . 13 (𝑤 ∈ ℝ → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1615adantr 479 . . . . . . . . . . . 12 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ ∃𝑥𝐴 -𝑤 = 𝐵))
1712, 16mpbid 231 . . . . . . . . . . 11 ((𝑤 ∈ ℝ ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 -𝑤 = 𝐵)
1817adantll 712 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 -𝑤 = 𝐵)
19 nfv 1909 . . . . . . . . . . . . 13 𝑥 𝑤 ∈ ℝ
201, 19nfan 1894 . . . . . . . . . . . 12 𝑥(𝜑𝑤 ∈ ℝ)
21 negeq 11477 . . . . . . . . . . . . . . . . . . 19 (-𝑤 = 𝐵 → --𝑤 = -𝐵)
2221eqcomd 2731 . . . . . . . . . . . . . . . . . 18 (-𝑤 = 𝐵 → -𝐵 = --𝑤)
2322adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → -𝐵 = --𝑤)
24 recn 11223 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ ℝ → 𝑤 ∈ ℂ)
2524negnegd 11587 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ℝ → --𝑤 = 𝑤)
2625adantr 479 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → --𝑤 = 𝑤)
2723, 26eqtr2d 2766 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ ℝ ∧ -𝑤 = 𝐵) → 𝑤 = -𝐵)
2827ex 411 . . . . . . . . . . . . . . 15 (𝑤 ∈ ℝ → (-𝑤 = 𝐵𝑤 = -𝐵))
2928adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ℝ) → (-𝑤 = 𝐵𝑤 = -𝐵))
3029adantr 479 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (-𝑤 = 𝐵𝑤 = -𝐵))
31 negeq 11477 . . . . . . . . . . . . . . . . 17 (𝑤 = -𝐵 → -𝑤 = --𝐵)
3231adantl 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = --𝐵)
333recnd 11267 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)
3433negnegd 11587 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → --𝐵 = 𝐵)
3534adantr 479 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → --𝐵 = 𝐵)
3632, 35eqtrd 2765 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑤 = -𝐵) → -𝑤 = 𝐵)
3736ex 411 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
3837adantlr 713 . . . . . . . . . . . . 13 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (𝑤 = -𝐵 → -𝑤 = 𝐵))
3930, 38impbid 211 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ℝ) ∧ 𝑥𝐴) → (-𝑤 = 𝐵𝑤 = -𝐵))
4020, 39rexbida 3260 . . . . . . . . . . 11 ((𝜑𝑤 ∈ ℝ) → (∃𝑥𝐴 -𝑤 = 𝐵 ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4140adantr 479 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → (∃𝑥𝐴 -𝑤 = 𝐵 ↔ ∃𝑥𝐴 𝑤 = -𝐵))
4218, 41mpbid 231 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → ∃𝑥𝐴 𝑤 = -𝐵)
43 simplr 767 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → 𝑤 ∈ ℝ)
4411, 42, 43elrnmptd 5958 . . . . . . . 8 (((𝜑𝑤 ∈ ℝ) ∧ -𝑤 ∈ ran (𝑥𝐴𝐵)) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵))
4544ex 411 . . . . . . 7 ((𝜑𝑤 ∈ ℝ) → (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
4645ralrimiva 3136 . . . . . 6 (𝜑 → ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
47 rabss 4062 . . . . . 6 ({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝐵) ↔ ∀𝑤 ∈ ℝ (-𝑤 ∈ ran (𝑥𝐴𝐵) → 𝑤 ∈ ran (𝑥𝐴 ↦ -𝐵)))
4846, 47sylibr 233 . . . . 5 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} ⊆ ran (𝑥𝐴 ↦ -𝐵))
49 nfcv 2892 . . . . . . . 8 𝑥-𝑤
50 nfmpt1 5252 . . . . . . . . 9 𝑥(𝑥𝐴𝐵)
5150nfrn 5949 . . . . . . . 8 𝑥ran (𝑥𝐴𝐵)
5249, 51nfel 2907 . . . . . . 7 𝑥-𝑤 ∈ ran (𝑥𝐴𝐵)
53 nfcv 2892 . . . . . . 7 𝑥
5452, 53nfrabw 3457 . . . . . 6 𝑥{𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}
5531eleq1d 2810 . . . . . . 7 (𝑤 = -𝐵 → (-𝑤 ∈ ran (𝑥𝐴𝐵) ↔ --𝐵 ∈ ran (𝑥𝐴𝐵)))
563renegcld 11666 . . . . . . 7 ((𝜑𝑥𝐴) → -𝐵 ∈ ℝ)
57 simpr 483 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
582elrnmpt1 5955 . . . . . . . . 9 ((𝑥𝐴𝐵 ∈ ℝ) → 𝐵 ∈ ran (𝑥𝐴𝐵))
5957, 3, 58syl2anc 582 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐵 ∈ ran (𝑥𝐴𝐵))
6034, 59eqeltrd 2825 . . . . . . 7 ((𝜑𝑥𝐴) → --𝐵 ∈ ran (𝑥𝐴𝐵))
6155, 56, 60elrabd 3678 . . . . . 6 ((𝜑𝑥𝐴) → -𝐵 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
621, 54, 11, 61rnmptssdf 44689 . . . . 5 (𝜑 → ran (𝑥𝐴 ↦ -𝐵) ⊆ {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)})
6348, 62eqssd 3991 . . . 4 (𝜑 → {𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)} = ran (𝑥𝐴 ↦ -𝐵))
6463infeq1d 9495 . . 3 (𝜑 → inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
6564negeqd 11479 . 2 (𝜑 → -inf({𝑤 ∈ ℝ ∣ -𝑤 ∈ ran (𝑥𝐴𝐵)}, ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
6610, 65eqtrd 2765 1 (𝜑 → sup(ran (𝑥𝐴𝐵), ℝ, < ) = -inf(ran (𝑥𝐴 ↦ -𝐵), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wnf 1777  wcel 2098  wne 2930  wral 3051  wrex 3060  {crab 3419  wss 3941  c0 4319   class class class wbr 5144  cmpt 5227  ran crn 5674  supcsup 9458  infcinf 9459  cr 11132   < clt 11273  cle 11274  -cneg 11470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-po 5585  df-so 5586  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9460  df-inf 9461  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472
This theorem is referenced by: (None)
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