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Theorem supminfxr2 44477
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 13193 . . . . . 6 -𝑒-∞ = +∞
21eqcomi 2739 . . . . 5 +∞ = -𝑒-∞
32a1i 11 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → +∞ = -𝑒-∞)
4 supminfxr2.1 . . . . 5 (𝜑𝐴 ⊆ ℝ*)
5 supxrpnf 13301 . . . . 5 ((𝐴 ⊆ ℝ* ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
64, 5sylan 578 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = +∞)
7 ssrab2 4076 . . . . . . . 8 {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ*
87a1i 11 . . . . . . 7 (+∞ ∈ 𝐴 → {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ⊆ ℝ*)
9 xnegeq 13190 . . . . . . . . . 10 (𝑦 = -∞ → -𝑒𝑦 = -𝑒-∞)
101a1i 11 . . . . . . . . . 10 (𝑦 = -∞ → -𝑒-∞ = +∞)
119, 10eqtrd 2770 . . . . . . . . 9 (𝑦 = -∞ → -𝑒𝑦 = +∞)
1211eleq1d 2816 . . . . . . . 8 (𝑦 = -∞ → (-𝑒𝑦𝐴 ↔ +∞ ∈ 𝐴))
13 mnfxr 11275 . . . . . . . . 9 -∞ ∈ ℝ*
1413a1i 11 . . . . . . . 8 (+∞ ∈ 𝐴 → -∞ ∈ ℝ*)
15 id 22 . . . . . . . 8 (+∞ ∈ 𝐴 → +∞ ∈ 𝐴)
1612, 14, 15elrabd 3684 . . . . . . 7 (+∞ ∈ 𝐴 → -∞ ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
17 infxrmnf 13320 . . . . . . 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 44445 . . . 4 ((𝜑 ∧ +∞ ∈ 𝐴) → -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = -𝑒-∞)
213, 6, 203eqtr4d 2780 . . 3 ((𝜑 ∧ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
224ssdifssd 4141 . . . . . . 7 (𝜑 → (𝐴 ∖ {-∞}) ⊆ ℝ*)
2322adantr 479 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ*)
24 difssd 4131 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → (𝐴 ∖ {-∞}) ⊆ 𝐴)
25 id 22 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ 𝐴)
26 ssnel 44029 . . . . . . . 8 (((𝐴 ∖ {-∞}) ⊆ 𝐴 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
2724, 25, 26syl2anc 582 . . . . . . 7 (¬ +∞ ∈ 𝐴 → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
2827adantl 480 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ +∞ ∈ (𝐴 ∖ {-∞}))
29 neldifsnd 4795 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → ¬ -∞ ∈ (𝐴 ∖ {-∞}))
3023, 28, 29xrssre 44356 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → (𝐴 ∖ {-∞}) ⊆ ℝ)
3130supminfxr 44472 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup((𝐴 ∖ {-∞}), ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
32 supxrmnf2 44441 . . . . . . 7 (𝐴 ⊆ ℝ* → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
334, 32syl 17 . . . . . 6 (𝜑 → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
3433eqcomd 2736 . . . . 5 (𝜑 → sup(𝐴, ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < ))
3534adantr 479 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = sup((𝐴 ∖ {-∞}), ℝ*, < ))
36 rexr 11264 . . . . . . . . . . . . . . . . 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 44133 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑒𝑦 = -𝑦)
40 eldifi 4125 . . . . . . . . . . . . . . . . . 18 (-𝑦 ∈ (𝐴 ∖ {-∞}) → -𝑦𝐴)
4140adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑦𝐴)
4239, 41eqeltrd 2831 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → -𝑒𝑦𝐴)
4337, 42jca 510 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → (𝑦 ∈ ℝ* ∧ -𝑒𝑦𝐴))
44 rabid 3450 . . . . . . . . . . . . . . 15 (𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ↔ (𝑦 ∈ ℝ* ∧ -𝑒𝑦𝐴))
4543, 44sylibr 233 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
46 renepnf 11266 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℝ → 𝑦 ≠ +∞)
47 elsni 4644 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {+∞} → 𝑦 = +∞)
4847necon3ai 2963 . . . . . . . . . . . . . . . 16 (𝑦 ≠ +∞ → ¬ 𝑦 ∈ {+∞})
4946, 48syl 17 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ → ¬ 𝑦 ∈ {+∞})
5038, 49syl 17 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → ¬ 𝑦 ∈ {+∞})
5145, 50eldifd 3958 . . . . . . . . . . . . 13 ((𝑦 ∈ ℝ ∧ -𝑦 ∈ (𝐴 ∖ {-∞})) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
5251ex 411 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})))
5352rgen 3061 . . . . . . . . . . 11 𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
5453a1i 11 . . . . . . . . . 10 (¬ +∞ ∈ 𝐴 → ∀𝑦 ∈ ℝ (-𝑦 ∈ (𝐴 ∖ {-∞}) → 𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})))
55 nfrab1 3449 . . . . . . . . . . . 12 𝑦{𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}
56 nfcv 2901 . . . . . . . . . . . 12 𝑦{+∞}
5755, 56nfdif 4124 . . . . . . . . . . 11 𝑦({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})
5857rabssf 44109 . . . . . . . . . 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 1915 . . . . . . . . . . . 12 𝑦 ¬ +∞ ∈ 𝐴
61 nfcv 2901 . . . . . . . . . . . 12 𝑦
62 eldifi 4125 . . . . . . . . . . . . . . 15 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ∈ {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴})
637, 62sselid 3979 . . . . . . . . . . . . . 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 710 . . . . . . . . . . . . . . . 16 (((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) ∧ 𝑦 = -∞) → +∞ ∈ 𝐴)
69 simpll 763 . . . . . . . . . . . . . . . 16 (((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) ∧ 𝑦 = -∞) → ¬ +∞ ∈ 𝐴)
7068, 69pm2.65da 813 . . . . . . . . . . . . . . 15 ((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) → ¬ 𝑦 = -∞)
7170neqned 2945 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴 ∧ -𝑒𝑦𝐴) → 𝑦 ≠ -∞)
7266, 71sylan2 591 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ≠ -∞)
73 eldifsni 4792 . . . . . . . . . . . . . 14 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → 𝑦 ≠ +∞)
7473adantl 480 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ≠ +∞)
7564, 72, 74xrred 44373 . . . . . . . . . . . 12 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → 𝑦 ∈ ℝ)
7660, 57, 61, 75ssdf2 44131 . . . . . . . . . . 11 (¬ +∞ ∈ 𝐴 → ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ⊆ ℝ)
7775rexnegd 44133 . . . . . . . . . . . . 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 4644 . . . . . . . . . . . . . . . . . 18 (-𝑒𝑦 ∈ {-∞} → -𝑒𝑦 = -∞)
8180adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → -𝑒𝑦 = -∞)
82 xnegeq 13190 . . . . . . . . . . . . . . . . . . . 20 (-𝑒𝑦 = -∞ → -𝑒-𝑒𝑦 = -𝑒-∞)
831a1i 11 . . . . . . . . . . . . . . . . . . . 20 (-𝑒𝑦 = -∞ → -𝑒-∞ = +∞)
8482, 83eqtr2d 2771 . . . . . . . . . . . . . . . . . . 19 (-𝑒𝑦 = -∞ → +∞ = -𝑒-𝑒𝑦)
8584adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → +∞ = -𝑒-𝑒𝑦)
86 xnegneg 13197 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ℝ* → -𝑒-𝑒𝑦 = 𝑦)
8786adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → -𝑒-𝑒𝑦 = 𝑦)
8885, 87eqtr2d 2771 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ℝ* ∧ -𝑒𝑦 = -∞) → 𝑦 = +∞)
8979, 81, 88syl2anc 582 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → 𝑦 = +∞)
9073neneqd 2943 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → ¬ 𝑦 = +∞)
9190adantr 479 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) ∧ -𝑒𝑦 ∈ {-∞}) → ¬ 𝑦 = +∞)
9289, 91pm2.65da 813 . . . . . . . . . . . . . . 15 (𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}) → ¬ -𝑒𝑦 ∈ {-∞})
9392adantl 480 . . . . . . . . . . . . . 14 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → ¬ -𝑒𝑦 ∈ {-∞})
9478, 93eldifd 3958 . . . . . . . . . . . . 13 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑒𝑦 ∈ (𝐴 ∖ {-∞}))
9577, 94eqeltrrd 2832 . . . . . . . . . . . 12 ((¬ +∞ ∈ 𝐴𝑦 ∈ ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞})) → -𝑦 ∈ (𝐴 ∖ {-∞}))
9695ralrimiva 3144 . . . . . . . . . . 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 44104 . . . . . . . . . 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 3998 . . . . . . . 8 (¬ +∞ ∈ 𝐴 → {𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})} = ({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}))
101100infeq1d 9474 . . . . . . 7 (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ) = inf(({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} ∖ {+∞}), ℝ*, < ))
102 infxrpnf2 44471 . . . . . . . . 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 2771 . . . . . 6 (¬ +∞ ∈ 𝐴 → inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = inf({𝑦 ∈ ℝ ∣ -𝑦 ∈ (𝐴 ∖ {-∞})}, ℝ*, < ))
106105xnegeqd 44445 . . . . 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 2780 . . 3 ((𝜑 ∧ ¬ +∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
10921, 108pm2.61dan 809 . 2 (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ))
110 xnegeq 13190 . . . . . . 7 (𝑦 = 𝑥 → -𝑒𝑦 = -𝑒𝑥)
111110eleq1d 2816 . . . . . 6 (𝑦 = 𝑥 → (-𝑒𝑦𝐴 ↔ -𝑒𝑥𝐴))
112111cbvrabv 3440 . . . . 5 {𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴} = {𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}
113112infeq1i 9475 . . . 4 inf({𝑦 ∈ ℝ* ∣ -𝑒𝑦𝐴}, ℝ*, < ) = inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥𝐴}, ℝ*, < )
114113xnegeqi 44448 . . 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 2770 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 1539  wcel 2104  wne 2938  wral 3059  {crab 3430  cdif 3944  wss 3947  {csn 4627  supcsup 9437  infcinf 9438  cr 11111  +∞cpnf 11249  -∞cmnf 11250  *cxr 11251   < clt 11252  -cneg 11449  -𝑒cxne 13093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-xneg 13096
This theorem is referenced by:  supminfxrrnmpt  44479  liminfvalxr  44797
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