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Theorem supminfxr2 43694
Description: The extended real suprema of a set of extended reals is the extended real negative of the extended real infima of that set's image under extended real negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
supminfxr2.1 (𝜑𝐴 ⊆ ℝ*)
Assertion
Ref Expression
supminfxr2 (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < ))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem supminfxr2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 xnegmnf 13129 . . . . . 6 -𝑒-∞ = +∞
21eqcomi 2745 . . . . 5 +∞ = -𝑒-∞
32a1i 11 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → +∞ = -𝑒-∞)
4 supminfxr2.1 . . . . 5 (𝜑𝐴 ⊆ ℝ*)
5 supxrpnf 13237 . . . . 5 ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
64, 5sylan 580 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
7 ssrab2 4037 . . . . . . . 8 {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ*
87a1i 11 . . . . . . 7 (+∞ ∈ 𝐴 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ*)
9 xnegeq 13126 . . . . . . . . . 10 (𝑦 = -∞ → -𝑒𝑦 = -𝑒-∞)
101a1i 11 . . . . . . . . . 10 (𝑦 = -∞ → -𝑒-∞ = +∞)
119, 10eqtrd 2776 . . . . . . . . 9 (𝑦 = -∞ → -𝑒𝑦 = +∞)
1211eleq1d 2822 . . . . . . . 8 (𝑦 = -∞ → (-𝑒𝑦𝐴 ↔ +∞ ∈ 𝐴))
13 mnfxr 11212 . . . . . . . . 9 -∞ ∈ ℝ*
1413a1i 11 . . . . . . . 8 (+∞ ∈ 𝐴 → -∞ ∈ ℝ*)
15 id 22 . . . . . . . 8 (+∞ ∈ 𝐴 → +∞ ∈ 𝐴)
1612, 14, 15elrabd 3647 . . . . . . 7 (+∞ ∈ 𝐴 → -∞ ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
17 infxrmnf 13256 . . . . . . 7 (({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ* ∧ -∞ ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}) → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -∞)
188, 16, 17syl2anc 584 . . . . . 6 (+∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -∞)
1918adantl 482 . . . . 5 ((𝜑 ∧ +∞ ∈ 𝐴) → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -∞)
2019xnegeqd 43662 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒-∞)
213, 6, 203eqtr4d 2786 . . 3 ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
224ssdifssd 4102 . . . . . . 7 (𝜑 → (𝐴 ∖ {-∞}) ⊆ ℝ*)
2322adantr 481 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ*)
24 difssd 4092 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) ⊆ 𝐴)
25 id 22 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ 𝐴)
26 ssnel 43238 . . . . . . . 8 (((𝐴 ∖ {-∞}) ⊆ 𝐴 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
2724, 25, 26syl2anc 584 . . . . . . 7 (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
2827adantl 482 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
29 neldifsnd 4753 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ -∞ ∈ (𝐴 ∖ {-∞}))
3023, 28, 29xrssre 43572 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ)
3130supminfxr 43689 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
32 supxrmnf2 43658 . . . . . . 7 (𝐴 ⊆ ℝ* → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
334, 32syl 17 . . . . . 6 (𝜑 → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
3433eqcomd 2742 . . . . 5 (𝜑 → sup(𝐴, ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < ))
3534adantr 481 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < ))
36 rexr 11201 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
3736adantr 481 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ*)
38 simpl 483 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ)
3938rexnegd 43343 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑒𝑦 = -𝑦)
40 eldifi 4086 . . . . . . . . . . . . . . . . . 18 (-𝑦 ∈ (𝐴 ∖ {-∞}) → -𝑦𝐴)
4140adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑦𝐴)
4239, 41eqeltrd 2838 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑒𝑦𝐴)
4337, 42jca 512 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → (𝑦 ∈ ℝ* ∧ -𝑒𝑦𝐴))
44 rabid 3427 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ↔ (𝑦 ∈ ℝ* ∧ -𝑒𝑦𝐴))
4543, 44sylibr 233 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
46 renepnf 11203 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ → 𝑦 ≠ +∞)
47 elsni 4603 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {+∞} → 𝑦 = +∞)
4847necon3ai 2968 . . . . . . . . . . . . . . . 16 (𝑦 ≠ +∞ → ¬ 𝑦 ∈ {+∞})
4946, 48syl 17 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ → ¬ 𝑦 ∈ {+∞})
5038, 49syl 17 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → ¬ 𝑦 ∈ {+∞})
5145, 50eldifd 3921 . . . . . . . . . . . . 13 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
5251ex 413 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})))
5352rgen 3066 . . . . . . . . . . 11 𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
5453a1i 11 . . . . . . . . . 10 (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})))
55 nfrab1 3426 . . . . . . . . . . . 12 𝑦{𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}
56 nfcv 2907 . . . . . . . . . . . 12 𝑦{+∞}
5755, 56nfdif 4085 . . . . . . . . . . 11 𝑦({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})
5857rabssf 43319 . . . . . . . . . 10 ({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ⊆ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ↔ ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})))
5954, 58sylibr 233 . . . . . . . . 9 (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ⊆ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
60 nfv 1917 . . . . . . . . . . . 12 𝑦 ¬ +∞ ∈ 𝐴
61 nfcv 2907 . . . . . . . . . . . 12 𝑦
62 eldifi 4086 . . . . . . . . . . . . . . 15 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
637, 62sselid 3942 . . . . . . . . . . . . . 14 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ∈ ℝ*)
6463adantl 482 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ*)
6544simprbi 497 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} → -𝑒𝑦𝐴)
6662, 65syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → -𝑒𝑦𝐴)
6712biimpac 479 . . . . . . . . . . . . . . . . 17 ((-𝑒𝑦𝐴𝑦 = -∞) → +∞ ∈ 𝐴)
6867adantll 712 . . . . . . . . . . . . . . . 16 (((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
69 simpll 765 . . . . . . . . . . . . . . . 16 (((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) ∧ 𝑦 = -∞) → ¬ +∞ ∈ 𝐴)
7068, 69pm2.65da 815 . . . . . . . . . . . . . . 15 ((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) → ¬ 𝑦 = -∞)
7170neqned 2950 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) → 𝑦 ≠ -∞)
7266, 71sylan2 593 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ≠ -∞)
73 eldifsni 4750 . . . . . . . . . . . . . 14 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ≠ +∞)
7473adantl 482 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ≠ +∞)
7564, 72, 74xrred 43589 . . . . . . . . . . . 12 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ)
7660, 57, 61, 75ssdf2 43341 . . . . . . . . . . 11 (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ⊆ ℝ)
7775rexnegd 43343 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑒𝑦 = -𝑦)
7866adantl 482 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑒𝑦𝐴)
7963adantr 481 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → 𝑦 ∈ ℝ*)
80 elsni 4603 . . . . . . . . . . . . . . . . . 18 (-𝑒𝑦 ∈ {-∞} → -𝑒𝑦 = -∞)
8180adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → -𝑒𝑦 = -∞)
82 xnegeq 13126 . . . . . . . . . . . . . . . . . . . 20 (-𝑒𝑦 = -∞ → -𝑒-𝑒𝑦 = -𝑒-∞)
831a1i 11 . . . . . . . . . . . . . . . . . . . 20 (-𝑒𝑦 = -∞ → -𝑒-∞ = +∞)
8482, 83eqtr2d 2777 . . . . . . . . . . . . . . . . . . 19 (-𝑒𝑦 = -∞ → +∞ = -𝑒-𝑒𝑦)
8584adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → +∞ = -𝑒-𝑒𝑦)
86 xnegneg 13133 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
8786adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → -𝑒-𝑒𝑦 = 𝑦)
8885, 87eqtr2d 2777 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → 𝑦 = +∞)
8979, 81, 88syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → 𝑦 = +∞)
9073neneqd 2948 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → ¬ 𝑦 = +∞)
9190adantr 481 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → ¬ 𝑦 = +∞)
9289, 91pm2.65da 815 . . . . . . . . . . . . . . 15 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → ¬ -𝑒𝑦 ∈ {-∞})
9392adantl 482 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → ¬ -𝑒𝑦 ∈ {-∞})
9478, 93eldifd 3921 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑒𝑦 ∈ (𝐴 ∖ {-∞}))
9577, 94eqeltrrd 2839 . . . . . . . . . . . 12 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑦 ∈ (𝐴 ∖ {-∞}))
9695ralrimiva 3143 . . . . . . . . . . 11 (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞}))
9776, 96jca 512 . . . . . . . . . 10 (¬ +∞ ∈ 𝐴 → (({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ⊆ ℝ ∧ ∀𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞})))
9857, 61ssrabf 43314 . . . . . . . . . 10 (({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ⊆ {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} ↔ (({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ⊆ ℝ ∧ ∀𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞})))
9997, 98sylibr 233 . . . . . . . . 9 (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ⊆ {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})})
10059, 99eqssd 3961 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} = ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
101100infeq1d 9413 . . . . . . 7 (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ) = inf(({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}), ℝ*, < ))
102 infxrpnf2 43688 . . . . . . . . 9 ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ* → inf(({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}), ℝ*, < ) = inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
1037, 102ax-mp 5 . . . . . . . 8 inf(({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}), ℝ*, < ) = inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < )
104103a1i 11 . . . . . . 7 (¬ +∞ ∈ 𝐴 → inf(({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}), ℝ*, < ) = inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
105101, 104eqtr2d 2777 . . . . . 6 (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
106105xnegeqd 43662 . . . . 5 (¬ +∞ ∈ 𝐴 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
107106adantl 482 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
10831, 35, 1073eqtr4d 2786 . . 3 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
10921, 108pm2.61dan 811 . 2 (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
110 xnegeq 13126 . . . . . . 7 (𝑦 = 𝑥 → -𝑒𝑦 = -𝑒𝑥)
111110eleq1d 2822 . . . . . 6 (𝑦 = 𝑥 → (-𝑒𝑦𝐴 ↔ -𝑒𝑥𝐴))
112111cbvrabv 3417 . . . . 5 {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} = {𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}
113112infeq1i 9414 . . . 4 inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < )
114113xnegeqi 43665 . . 3 -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < )
115114a1i 11 . 2 (𝜑 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < ))
116109, 115eqtrd 2776 1 (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  cdif 3907  wss 3910  {csn 4586  supcsup 9376  infcinf 9377  cr 11050  +∞cpnf 11186  -∞cmnf 11187  *cxr 11188   < clt 11189  -cneg 11386  -𝑒cxne 13030
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-xneg 13033
This theorem is referenced by:  supminfxrrnmpt  43696  liminfvalxr  44014
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