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Theorem supminfxr2 44478
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 13194 . . . . . 6 -𝑒-∞ = +∞
21eqcomi 2740 . . . . 5 +∞ = -𝑒-∞
32a1i 11 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → +∞ = -𝑒-∞)
4 supminfxr2.1 . . . . 5 (𝜑𝐴 ⊆ ℝ*)
5 supxrpnf 13302 . . . . 5 ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
64, 5sylan 579 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
7 ssrab2 4077 . . . . . . . 8 {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ*
87a1i 11 . . . . . . 7 (+∞ ∈ 𝐴 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ*)
9 xnegeq 13191 . . . . . . . . . 10 (𝑦 = -∞ → -𝑒𝑦 = -𝑒-∞)
101a1i 11 . . . . . . . . . 10 (𝑦 = -∞ → -𝑒-∞ = +∞)
119, 10eqtrd 2771 . . . . . . . . 9 (𝑦 = -∞ → -𝑒𝑦 = +∞)
1211eleq1d 2817 . . . . . . . 8 (𝑦 = -∞ → (-𝑒𝑦𝐴 ↔ +∞ ∈ 𝐴))
13 mnfxr 11276 . . . . . . . . 9 -∞ ∈ ℝ*
1413a1i 11 . . . . . . . 8 (+∞ ∈ 𝐴 → -∞ ∈ ℝ*)
15 id 22 . . . . . . . 8 (+∞ ∈ 𝐴 → +∞ ∈ 𝐴)
1612, 14, 15elrabd 3685 . . . . . . 7 (+∞ ∈ 𝐴 → -∞ ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
17 infxrmnf 13321 . . . . . . 7 (({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ* ∧ -∞ ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}) → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -∞)
188, 16, 17syl2anc 583 . . . . . 6 (+∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -∞)
1918adantl 481 . . . . 5 ((𝜑 ∧ +∞ ∈ 𝐴) → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -∞)
2019xnegeqd 44446 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒-∞)
213, 6, 203eqtr4d 2781 . . 3 ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
224ssdifssd 4142 . . . . . . 7 (𝜑 → (𝐴 ∖ {-∞}) ⊆ ℝ*)
2322adantr 480 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ*)
24 difssd 4132 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) ⊆ 𝐴)
25 id 22 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ 𝐴)
26 ssnel 44030 . . . . . . . 8 (((𝐴 ∖ {-∞}) ⊆ 𝐴 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
2724, 25, 26syl2anc 583 . . . . . . 7 (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
2827adantl 481 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
29 neldifsnd 4796 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ -∞ ∈ (𝐴 ∖ {-∞}))
3023, 28, 29xrssre 44357 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ)
3130supminfxr 44473 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
32 supxrmnf2 44442 . . . . . . 7 (𝐴 ⊆ ℝ* → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
334, 32syl 17 . . . . . 6 (𝜑 → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
3433eqcomd 2737 . . . . 5 (𝜑 → sup(𝐴, ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < ))
3534adantr 480 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < ))
36 rexr 11265 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
3736adantr 480 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ*)
38 simpl 482 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ℝ)
3938rexnegd 44134 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑒𝑦 = -𝑦)
40 eldifi 4126 . . . . . . . . . . . . . . . . . 18 (-𝑦 ∈ (𝐴 ∖ {-∞}) → -𝑦𝐴)
4140adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑦𝐴)
4239, 41eqeltrd 2832 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑒𝑦𝐴)
4337, 42jca 511 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → (𝑦 ∈ ℝ* ∧ -𝑒𝑦𝐴))
44 rabid 3451 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ↔ (𝑦 ∈ ℝ* ∧ -𝑒𝑦𝐴))
4543, 44sylibr 233 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
46 renepnf 11267 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ → 𝑦 ≠ +∞)
47 elsni 4645 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {+∞} → 𝑦 = +∞)
4847necon3ai 2964 . . . . . . . . . . . . . . . 16 (𝑦 ≠ +∞ → ¬ 𝑦 ∈ {+∞})
4946, 48syl 17 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ → ¬ 𝑦 ∈ {+∞})
5038, 49syl 17 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → ¬ 𝑦 ∈ {+∞})
5145, 50eldifd 3959 . . . . . . . . . . . . 13 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
5251ex 412 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})))
5352rgen 3062 . . . . . . . . . . 11 𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
5453a1i 11 . . . . . . . . . 10 (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})))
55 nfrab1 3450 . . . . . . . . . . . 12 𝑦{𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}
56 nfcv 2902 . . . . . . . . . . . 12 𝑦{+∞}
5755, 56nfdif 4125 . . . . . . . . . . 11 𝑦({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})
5857rabssf 44110 . . . . . . . . . 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 1916 . . . . . . . . . . . 12 𝑦 ¬ +∞ ∈ 𝐴
61 nfcv 2902 . . . . . . . . . . . 12 𝑦
62 eldifi 4126 . . . . . . . . . . . . . . 15 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
637, 62sselid 3980 . . . . . . . . . . . . . 14 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ∈ ℝ*)
6463adantl 481 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ*)
6544simprbi 496 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} → -𝑒𝑦𝐴)
6662, 65syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → -𝑒𝑦𝐴)
6712biimpac 478 . . . . . . . . . . . . . . . . 17 ((-𝑒𝑦𝐴𝑦 = -∞) → +∞ ∈ 𝐴)
6867adantll 711 . . . . . . . . . . . . . . . 16 (((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
69 simpll 764 . . . . . . . . . . . . . . . 16 (((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) ∧ 𝑦 = -∞) → ¬ +∞ ∈ 𝐴)
7068, 69pm2.65da 814 . . . . . . . . . . . . . . 15 ((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) → ¬ 𝑦 = -∞)
7170neqned 2946 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) → 𝑦 ≠ -∞)
7266, 71sylan2 592 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ≠ -∞)
73 eldifsni 4793 . . . . . . . . . . . . . 14 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ≠ +∞)
7473adantl 481 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ≠ +∞)
7564, 72, 74xrred 44374 . . . . . . . . . . . 12 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ)
7660, 57, 61, 75ssdf2 44132 . . . . . . . . . . 11 (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ⊆ ℝ)
7775rexnegd 44134 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑒𝑦 = -𝑦)
7866adantl 481 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑒𝑦𝐴)
7963adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → 𝑦 ∈ ℝ*)
80 elsni 4645 . . . . . . . . . . . . . . . . . 18 (-𝑒𝑦 ∈ {-∞} → -𝑒𝑦 = -∞)
8180adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → -𝑒𝑦 = -∞)
82 xnegeq 13191 . . . . . . . . . . . . . . . . . . . 20 (-𝑒𝑦 = -∞ → -𝑒-𝑒𝑦 = -𝑒-∞)
831a1i 11 . . . . . . . . . . . . . . . . . . . 20 (-𝑒𝑦 = -∞ → -𝑒-∞ = +∞)
8482, 83eqtr2d 2772 . . . . . . . . . . . . . . . . . . 19 (-𝑒𝑦 = -∞ → +∞ = -𝑒-𝑒𝑦)
8584adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → +∞ = -𝑒-𝑒𝑦)
86 xnegneg 13198 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
8786adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → -𝑒-𝑒𝑦 = 𝑦)
8885, 87eqtr2d 2772 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → 𝑦 = +∞)
8979, 81, 88syl2anc 583 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → 𝑦 = +∞)
9073neneqd 2944 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → ¬ 𝑦 = +∞)
9190adantr 480 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → ¬ 𝑦 = +∞)
9289, 91pm2.65da 814 . . . . . . . . . . . . . . 15 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → ¬ -𝑒𝑦 ∈ {-∞})
9392adantl 481 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → ¬ -𝑒𝑦 ∈ {-∞})
9478, 93eldifd 3959 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑒𝑦 ∈ (𝐴 ∖ {-∞}))
9577, 94eqeltrrd 2833 . . . . . . . . . . . 12 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑦 ∈ (𝐴 ∖ {-∞}))
9695ralrimiva 3145 . . . . . . . . . . 11 (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞}))
9776, 96jca 511 . . . . . . . . . 10 (¬ +∞ ∈ 𝐴 → (({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ⊆ ℝ ∧ ∀𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})-𝑦 ∈ (𝐴 ∖ {-∞})))
9857, 61ssrabf 44105 . . . . . . . . . 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 3999 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} = ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
101100infeq1d 9476 . . . . . . 7 (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ) = inf(({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}), ℝ*, < ))
102 infxrpnf2 44472 . . . . . . . . 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 2772 . . . . . 6 (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
106105xnegeqd 44446 . . . . 5 (¬ +∞ ∈ 𝐴 → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
107106adantl 481 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
10831, 35, 1073eqtr4d 2781 . . 3 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
10921, 108pm2.61dan 810 . 2 (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
110 xnegeq 13191 . . . . . . 7 (𝑦 = 𝑥 → -𝑒𝑦 = -𝑒𝑥)
111110eleq1d 2817 . . . . . 6 (𝑦 = 𝑥 → (-𝑒𝑦𝐴 ↔ -𝑒𝑥𝐴))
112111cbvrabv 3441 . . . . 5 {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} = {𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}
113112infeq1i 9477 . . . 4 inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < )
114113xnegeqi 44449 . . 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 2771 1 (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2105  wne 2939  wral 3060  {crab 3431  cdif 3945  wss 3948  {csn 4628  supcsup 9439  infcinf 9440  cr 11113  +∞cpnf 11250  -∞cmnf 11251  *cxr 11252   < clt 11253  -cneg 11450  -𝑒cxne 13094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-sup 9441  df-inf 9442  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-xneg 13097
This theorem is referenced by:  supminfxrrnmpt  44480  liminfvalxr  44798
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