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Theorem supminfxr2 44989
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 13224 . . . . . 6 -𝑒-∞ = +∞
21eqcomi 2734 . . . . 5 +∞ = -𝑒-∞
32a1i 11 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → +∞ = -𝑒-∞)
4 supminfxr2.1 . . . . 5 (𝜑𝐴 ⊆ ℝ*)
5 supxrpnf 13332 . . . . 5 ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
64, 5sylan 578 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
7 ssrab2 4073 . . . . . . . 8 {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ*
87a1i 11 . . . . . . 7 (+∞ ∈ 𝐴 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ*)
9 xnegeq 13221 . . . . . . . . . 10 (𝑦 = -∞ → -𝑒𝑦 = -𝑒-∞)
101a1i 11 . . . . . . . . . 10 (𝑦 = -∞ → -𝑒-∞ = +∞)
119, 10eqtrd 2765 . . . . . . . . 9 (𝑦 = -∞ → -𝑒𝑦 = +∞)
1211eleq1d 2810 . . . . . . . 8 (𝑦 = -∞ → (-𝑒𝑦𝐴 ↔ +∞ ∈ 𝐴))
13 mnfxr 11303 . . . . . . . . 9 -∞ ∈ ℝ*
1413a1i 11 . . . . . . . 8 (+∞ ∈ 𝐴 → -∞ ∈ ℝ*)
15 id 22 . . . . . . . 8 (+∞ ∈ 𝐴 → +∞ ∈ 𝐴)
1612, 14, 15elrabd 3681 . . . . . . 7 (+∞ ∈ 𝐴 → -∞ ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
17 infxrmnf 13351 . . . . . . 7 (({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ* ∧ -∞ ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}) → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -∞)
188, 16, 17syl2anc 582 . . . . . 6 (+∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -∞)
1918adantl 480 . . . . 5 ((𝜑 ∧ +∞ ∈ 𝐴) → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -∞)
2019xnegeqd 44957 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒-∞)
213, 6, 203eqtr4d 2775 . . 3 ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
224ssdifssd 4139 . . . . . . 7 (𝜑 → (𝐴 ∖ {-∞}) ⊆ ℝ*)
2322adantr 479 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ*)
24 difssd 4129 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) ⊆ 𝐴)
25 id 22 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ 𝐴)
26 ssnel 44547 . . . . . . . 8 (((𝐴 ∖ {-∞}) ⊆ 𝐴 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
2724, 25, 26syl2anc 582 . . . . . . 7 (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
2827adantl 480 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
29 neldifsnd 4798 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ -∞ ∈ (𝐴 ∖ {-∞}))
3023, 28, 29xrssre 44868 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ)
3130supminfxr 44984 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
32 supxrmnf2 44953 . . . . . . 7 (𝐴 ⊆ ℝ* → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
334, 32syl 17 . . . . . 6 (𝜑 → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
3433eqcomd 2731 . . . . 5 (𝜑 → sup(𝐴, ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < ))
3534adantr 479 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < ))
36 rexr 11292 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
3736adantr 479 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ*)
38 simpl 481 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ)
3938rexnegd 44649 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑒𝑦 = -𝑦)
40 eldifi 4123 . . . . . . . . . . . . . . . . . 18 (-𝑦 ∈ (𝐴 ∖ {-∞}) → -𝑦𝐴)
4140adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑦𝐴)
4239, 41eqeltrd 2825 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑒𝑦𝐴)
4337, 42jca 510 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → (𝑦 ∈ ℝ* ∧ -𝑒𝑦𝐴))
44 rabid 3439 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ↔ (𝑦 ∈ ℝ* ∧ -𝑒𝑦𝐴))
4543, 44sylibr 233 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
46 renepnf 11294 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ → 𝑦 ≠ +∞)
47 elsni 4647 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {+∞} → 𝑦 = +∞)
4847necon3ai 2954 . . . . . . . . . . . . . . . 16 (𝑦 ≠ +∞ → ¬ 𝑦 ∈ {+∞})
4946, 48syl 17 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ → ¬ 𝑦 ∈ {+∞})
5038, 49syl 17 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → ¬ 𝑦 ∈ {+∞})
5145, 50eldifd 3955 . . . . . . . . . . . . 13 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
5251ex 411 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})))
5352rgen 3052 . . . . . . . . . . 11 𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
5453a1i 11 . . . . . . . . . 10 (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})))
55 nfrab1 3438 . . . . . . . . . . . 12 𝑦{𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}
56 nfcv 2891 . . . . . . . . . . . 12 𝑦{+∞}
5755, 56nfdif 4121 . . . . . . . . . . 11 𝑦({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})
5857rabssf 44625 . . . . . . . . . 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 1909 . . . . . . . . . . . 12 𝑦 ¬ +∞ ∈ 𝐴
61 nfcv 2891 . . . . . . . . . . . 12 𝑦
62 eldifi 4123 . . . . . . . . . . . . . . 15 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
637, 62sselid 3974 . . . . . . . . . . . . . 14 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ∈ ℝ*)
6463adantl 480 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ*)
6544simprbi 495 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} → -𝑒𝑦𝐴)
6662, 65syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → -𝑒𝑦𝐴)
6712biimpac 477 . . . . . . . . . . . . . . . . 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 2936 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) → 𝑦 ≠ -∞)
7266, 71sylan2 591 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ≠ -∞)
73 eldifsni 4795 . . . . . . . . . . . . . 14 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ≠ +∞)
7473adantl 480 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ≠ +∞)
7564, 72, 74xrred 44885 . . . . . . . . . . . 12 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ)
7660, 57, 61, 75ssdf2 44647 . . . . . . . . . . 11 (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ⊆ ℝ)
7775rexnegd 44649 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑒𝑦 = -𝑦)
7866adantl 480 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑒𝑦𝐴)
7963adantr 479 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → 𝑦 ∈ ℝ*)
80 elsni 4647 . . . . . . . . . . . . . . . . . 18 (-𝑒𝑦 ∈ {-∞} → -𝑒𝑦 = -∞)
8180adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → -𝑒𝑦 = -∞)
82 xnegeq 13221 . . . . . . . . . . . . . . . . . . . 20 (-𝑒𝑦 = -∞ → -𝑒-𝑒𝑦 = -𝑒-∞)
831a1i 11 . . . . . . . . . . . . . . . . . . . 20 (-𝑒𝑦 = -∞ → -𝑒-∞ = +∞)
8482, 83eqtr2d 2766 . . . . . . . . . . . . . . . . . . 19 (-𝑒𝑦 = -∞ → +∞ = -𝑒-𝑒𝑦)
8584adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → +∞ = -𝑒-𝑒𝑦)
86 xnegneg 13228 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
8786adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → -𝑒-𝑒𝑦 = 𝑦)
8885, 87eqtr2d 2766 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → 𝑦 = +∞)
8979, 81, 88syl2anc 582 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → 𝑦 = +∞)
9073neneqd 2934 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → ¬ 𝑦 = +∞)
9190adantr 479 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → ¬ 𝑦 = +∞)
9289, 91pm2.65da 815 . . . . . . . . . . . . . . 15 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → ¬ -𝑒𝑦 ∈ {-∞})
9392adantl 480 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → ¬ -𝑒𝑦 ∈ {-∞})
9478, 93eldifd 3955 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑒𝑦 ∈ (𝐴 ∖ {-∞}))
9577, 94eqeltrrd 2826 . . . . . . . . . . . 12 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑦 ∈ (𝐴 ∖ {-∞}))
9695ralrimiva 3135 . . . . . . . . . . 11 (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞}))
9776, 96jca 510 . . . . . . . . . 10 (¬ +∞ ∈ 𝐴 → (({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ⊆ ℝ ∧ ∀𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞})))
9857, 61ssrabf 44620 . . . . . . . . . 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 3994 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} = ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
101100infeq1d 9502 . . . . . . 7 (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ) = inf(({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}), ℝ*, < ))
102 infxrpnf2 44983 . . . . . . . . 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 2766 . . . . . 6 (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
106105xnegeqd 44957 . . . . 5 (¬ +∞ ∈ 𝐴 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
107106adantl 480 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
10831, 35, 1073eqtr4d 2775 . . 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 13221 . . . . . . 7 (𝑦 = 𝑥 → -𝑒𝑦 = -𝑒𝑥)
111110eleq1d 2810 . . . . . 6 (𝑦 = 𝑥 → (-𝑒𝑦𝐴 ↔ -𝑒𝑥𝐴))
112111cbvrabv 3429 . . . . 5 {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} = {𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}
113112infeq1i 9503 . . . 4 inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < )
114113xnegeqi 44960 . . 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 2765 1 (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2929  wral 3050  {crab 3418  cdif 3941  wss 3944  {csn 4630  supcsup 9465  infcinf 9466  cr 11139  +∞cpnf 11277  -∞cmnf 11278  *cxr 11279   < clt 11280  -cneg 11477  -𝑒cxne 13124
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 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
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 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-xneg 13127
This theorem is referenced by:  supminfxrrnmpt  44991  liminfvalxr  45309
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