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 Description: The supremum function distributes over addition in a sense similar to that in supmul 10872. (Contributed by Brendan Leahy, 26-Sep-2017.)
Hypotheses
Ref Expression
supadd.a1 (𝜑𝐴 ⊆ ℝ)
supadd.a2 (𝜑𝐴 ≠ ∅)
supadd.a3 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)
supadd.b1 (𝜑𝐵 ⊆ ℝ)
supadd.b2 (𝜑𝐵 ≠ ∅)
supadd.b3 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥)
supadd.c 𝐶 = {𝑧 ∣ ∃𝑣𝐴𝑏𝐵 𝑧 = (𝑣 + 𝑏)}
Assertion
Ref Expression
supadd (𝜑 → (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < ))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑏,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑏,𝑣   𝑥,𝐶   𝜑,𝑧,𝑏,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑦,𝑧,𝑣,𝑏)

Proof of Theorem supadd
Dummy variables 𝑤 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supadd.a1 . . . . 5 (𝜑𝐴 ⊆ ℝ)
2 supadd.a2 . . . . 5 (𝜑𝐴 ≠ ∅)
3 supadd.a3 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)
4 supadd.b1 . . . . . 6 (𝜑𝐵 ⊆ ℝ)
5 supadd.b2 . . . . . 6 (𝜑𝐵 ≠ ∅)
6 supadd.b3 . . . . . 6 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥)
7 suprcl 10862 . . . . . 6 ((𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥) → sup(𝐵, ℝ, < ) ∈ ℝ)
84, 5, 6, 7syl3anc 1318 . . . . 5 (𝜑 → sup(𝐵, ℝ, < ) ∈ ℝ)
9 eqid 2610 . . . . 5 {𝑧 ∣ ∃𝑎𝐴 𝑧 = (𝑎 + sup(𝐵, ℝ, < ))} = {𝑧 ∣ ∃𝑎𝐴 𝑧 = (𝑎 + sup(𝐵, ℝ, < ))}
101, 2, 3, 8, 9supaddc 10867 . . . 4 (𝜑 → (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) = sup({𝑧 ∣ ∃𝑎𝐴 𝑧 = (𝑎 + sup(𝐵, ℝ, < ))}, ℝ, < ))
111sselda 3568 . . . . . . . . . 10 ((𝜑𝑎𝐴) → 𝑎 ∈ ℝ)
1211recnd 9947 . . . . . . . . 9 ((𝜑𝑎𝐴) → 𝑎 ∈ ℂ)
138adantr 480 . . . . . . . . . 10 ((𝜑𝑎𝐴) → sup(𝐵, ℝ, < ) ∈ ℝ)
1413recnd 9947 . . . . . . . . 9 ((𝜑𝑎𝐴) → sup(𝐵, ℝ, < ) ∈ ℂ)
1512, 14addcomd 10117 . . . . . . . 8 ((𝜑𝑎𝐴) → (𝑎 + sup(𝐵, ℝ, < )) = (sup(𝐵, ℝ, < ) + 𝑎))
1615eqeq2d 2620 . . . . . . 7 ((𝜑𝑎𝐴) → (𝑧 = (𝑎 + sup(𝐵, ℝ, < )) ↔ 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)))
1716rexbidva 3031 . . . . . 6 (𝜑 → (∃𝑎𝐴 𝑧 = (𝑎 + sup(𝐵, ℝ, < )) ↔ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)))
1817abbidv 2728 . . . . 5 (𝜑 → {𝑧 ∣ ∃𝑎𝐴 𝑧 = (𝑎 + sup(𝐵, ℝ, < ))} = {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)})
1918supeq1d 8235 . . . 4 (𝜑 → sup({𝑧 ∣ ∃𝑎𝐴 𝑧 = (𝑎 + sup(𝐵, ℝ, < ))}, ℝ, < ) = sup({𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}, ℝ, < ))
2010, 19eqtrd 2644 . . 3 (𝜑 → (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) = sup({𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}, ℝ, < ))
21 vex 3176 . . . . . . 7 𝑤 ∈ V
22 eqeq1 2614 . . . . . . . 8 (𝑧 = 𝑤 → (𝑧 = (sup(𝐵, ℝ, < ) + 𝑎) ↔ 𝑤 = (sup(𝐵, ℝ, < ) + 𝑎)))
2322rexbidv 3034 . . . . . . 7 (𝑧 = 𝑤 → (∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎) ↔ ∃𝑎𝐴 𝑤 = (sup(𝐵, ℝ, < ) + 𝑎)))
2421, 23elab 3319 . . . . . 6 (𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)} ↔ ∃𝑎𝐴 𝑤 = (sup(𝐵, ℝ, < ) + 𝑎))
254adantr 480 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → 𝐵 ⊆ ℝ)
265adantr 480 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → 𝐵 ≠ ∅)
276adantr 480 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥)
28 eqid 2610 . . . . . . . . . . 11 {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑏 + 𝑎)} = {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑏 + 𝑎)}
2925, 26, 27, 11, 28supaddc 10867 . . . . . . . . . 10 ((𝜑𝑎𝐴) → (sup(𝐵, ℝ, < ) + 𝑎) = sup({𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑏 + 𝑎)}, ℝ, < ))
304sselda 3568 . . . . . . . . . . . . . . . . 17 ((𝜑𝑏𝐵) → 𝑏 ∈ ℝ)
3130adantlr 747 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴) ∧ 𝑏𝐵) → 𝑏 ∈ ℝ)
3231recnd 9947 . . . . . . . . . . . . . . 15 (((𝜑𝑎𝐴) ∧ 𝑏𝐵) → 𝑏 ∈ ℂ)
3311adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑎𝐴) ∧ 𝑏𝐵) → 𝑎 ∈ ℝ)
3433recnd 9947 . . . . . . . . . . . . . . 15 (((𝜑𝑎𝐴) ∧ 𝑏𝐵) → 𝑎 ∈ ℂ)
3532, 34addcomd 10117 . . . . . . . . . . . . . 14 (((𝜑𝑎𝐴) ∧ 𝑏𝐵) → (𝑏 + 𝑎) = (𝑎 + 𝑏))
3635eqeq2d 2620 . . . . . . . . . . . . 13 (((𝜑𝑎𝐴) ∧ 𝑏𝐵) → (𝑧 = (𝑏 + 𝑎) ↔ 𝑧 = (𝑎 + 𝑏)))
3736rexbidva 3031 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → (∃𝑏𝐵 𝑧 = (𝑏 + 𝑎) ↔ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)))
3837abbidv 2728 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑏 + 𝑎)} = {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)})
3938supeq1d 8235 . . . . . . . . . 10 ((𝜑𝑎𝐴) → sup({𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑏 + 𝑎)}, ℝ, < ) = sup({𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}, ℝ, < ))
4029, 39eqtrd 2644 . . . . . . . . 9 ((𝜑𝑎𝐴) → (sup(𝐵, ℝ, < ) + 𝑎) = sup({𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}, ℝ, < ))
41 eqeq1 2614 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (𝑧 = (𝑎 + 𝑏) ↔ 𝑤 = (𝑎 + 𝑏)))
4241rexbidv 3034 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → (∃𝑏𝐵 𝑧 = (𝑎 + 𝑏) ↔ ∃𝑏𝐵 𝑤 = (𝑎 + 𝑏)))
4321, 42elab 3319 . . . . . . . . . . . 12 (𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)} ↔ ∃𝑏𝐵 𝑤 = (𝑎 + 𝑏))
44 rspe 2986 . . . . . . . . . . . . . . 15 ((𝑎𝐴 ∧ ∃𝑏𝐵 𝑤 = (𝑎 + 𝑏)) → ∃𝑎𝐴𝑏𝐵 𝑤 = (𝑎 + 𝑏))
45 oveq1 6556 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑎 → (𝑣 + 𝑏) = (𝑎 + 𝑏))
4645eqeq2d 2620 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑎 → (𝑧 = (𝑣 + 𝑏) ↔ 𝑧 = (𝑎 + 𝑏)))
4746rexbidv 3034 . . . . . . . . . . . . . . . . . 18 (𝑣 = 𝑎 → (∃𝑏𝐵 𝑧 = (𝑣 + 𝑏) ↔ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)))
4847cbvrexv 3148 . . . . . . . . . . . . . . . . 17 (∃𝑣𝐴𝑏𝐵 𝑧 = (𝑣 + 𝑏) ↔ ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎 + 𝑏))
49412rexbidv 3039 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑤 → (∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎 + 𝑏) ↔ ∃𝑎𝐴𝑏𝐵 𝑤 = (𝑎 + 𝑏)))
5048, 49syl5bb 271 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (∃𝑣𝐴𝑏𝐵 𝑧 = (𝑣 + 𝑏) ↔ ∃𝑎𝐴𝑏𝐵 𝑤 = (𝑎 + 𝑏)))
51 supadd.c . . . . . . . . . . . . . . . 16 𝐶 = {𝑧 ∣ ∃𝑣𝐴𝑏𝐵 𝑧 = (𝑣 + 𝑏)}
5221, 50, 51elab2 3323 . . . . . . . . . . . . . . 15 (𝑤𝐶 ↔ ∃𝑎𝐴𝑏𝐵 𝑤 = (𝑎 + 𝑏))
5344, 52sylibr 223 . . . . . . . . . . . . . 14 ((𝑎𝐴 ∧ ∃𝑏𝐵 𝑤 = (𝑎 + 𝑏)) → 𝑤𝐶)
5453ex 449 . . . . . . . . . . . . 13 (𝑎𝐴 → (∃𝑏𝐵 𝑤 = (𝑎 + 𝑏) → 𝑤𝐶))
551sseld 3567 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑎𝐴𝑎 ∈ ℝ))
564sseld 3567 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑏𝐵𝑏 ∈ ℝ))
5755, 56anim12d 584 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑎𝐴𝑏𝐵) → (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ)))
58 readdcl 9898 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎 + 𝑏) ∈ ℝ)
5957, 58syl6 34 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑎𝐴𝑏𝐵) → (𝑎 + 𝑏) ∈ ℝ))
60 eleq1a 2683 . . . . . . . . . . . . . . . . . 18 ((𝑎 + 𝑏) ∈ ℝ → (𝑤 = (𝑎 + 𝑏) → 𝑤 ∈ ℝ))
6159, 60syl6 34 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑎𝐴𝑏𝐵) → (𝑤 = (𝑎 + 𝑏) → 𝑤 ∈ ℝ)))
6261rexlimdvv 3019 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑎𝐴𝑏𝐵 𝑤 = (𝑎 + 𝑏) → 𝑤 ∈ ℝ))
6352, 62syl5bi 231 . . . . . . . . . . . . . . 15 (𝜑 → (𝑤𝐶𝑤 ∈ ℝ))
6463ssrdv 3574 . . . . . . . . . . . . . 14 (𝜑𝐶 ⊆ ℝ)
65 ovex 6577 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 + 𝑏) ∈ V
6665isseti 3182 . . . . . . . . . . . . . . . . . . . . 21 𝑤 𝑤 = (𝑎 + 𝑏)
6766rgenw 2908 . . . . . . . . . . . . . . . . . . . 20 𝑏𝐵𝑤 𝑤 = (𝑎 + 𝑏)
68 r19.2z 4012 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ≠ ∅ ∧ ∀𝑏𝐵𝑤 𝑤 = (𝑎 + 𝑏)) → ∃𝑏𝐵𝑤 𝑤 = (𝑎 + 𝑏))
695, 67, 68sylancl 693 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∃𝑏𝐵𝑤 𝑤 = (𝑎 + 𝑏))
70 rexcom4 3198 . . . . . . . . . . . . . . . . . . 19 (∃𝑏𝐵𝑤 𝑤 = (𝑎 + 𝑏) ↔ ∃𝑤𝑏𝐵 𝑤 = (𝑎 + 𝑏))
7169, 70sylib 207 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑤𝑏𝐵 𝑤 = (𝑎 + 𝑏))
7271ralrimivw 2950 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑎𝐴𝑤𝑏𝐵 𝑤 = (𝑎 + 𝑏))
73 r19.2z 4012 . . . . . . . . . . . . . . . . 17 ((𝐴 ≠ ∅ ∧ ∀𝑎𝐴𝑤𝑏𝐵 𝑤 = (𝑎 + 𝑏)) → ∃𝑎𝐴𝑤𝑏𝐵 𝑤 = (𝑎 + 𝑏))
742, 72, 73syl2anc 691 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑎𝐴𝑤𝑏𝐵 𝑤 = (𝑎 + 𝑏))
75 rexcom4 3198 . . . . . . . . . . . . . . . 16 (∃𝑎𝐴𝑤𝑏𝐵 𝑤 = (𝑎 + 𝑏) ↔ ∃𝑤𝑎𝐴𝑏𝐵 𝑤 = (𝑎 + 𝑏))
7674, 75sylib 207 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑤𝑎𝐴𝑏𝐵 𝑤 = (𝑎 + 𝑏))
77 n0 3890 . . . . . . . . . . . . . . . 16 (𝐶 ≠ ∅ ↔ ∃𝑤 𝑤𝐶)
7852exbii 1764 . . . . . . . . . . . . . . . 16 (∃𝑤 𝑤𝐶 ↔ ∃𝑤𝑎𝐴𝑏𝐵 𝑤 = (𝑎 + 𝑏))
7977, 78bitri 263 . . . . . . . . . . . . . . 15 (𝐶 ≠ ∅ ↔ ∃𝑤𝑎𝐴𝑏𝐵 𝑤 = (𝑎 + 𝑏))
8076, 79sylibr 223 . . . . . . . . . . . . . 14 (𝜑𝐶 ≠ ∅)
81 suprcl 10862 . . . . . . . . . . . . . . . . 17 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) → sup(𝐴, ℝ, < ) ∈ ℝ)
821, 2, 3, 81syl3anc 1318 . . . . . . . . . . . . . . . 16 (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
8382, 8readdcld 9948 . . . . . . . . . . . . . . 15 (𝜑 → (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) ∈ ℝ)
8411adantrr 749 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → 𝑎 ∈ ℝ)
8530adantrl 748 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → 𝑏 ∈ ℝ)
8682adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → sup(𝐴, ℝ, < ) ∈ ℝ)
878adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → sup(𝐵, ℝ, < ) ∈ ℝ)
881, 2, 33jca 1235 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥))
89 suprub 10863 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥) ∧ 𝑎𝐴) → 𝑎 ≤ sup(𝐴, ℝ, < ))
9088, 89sylan 487 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑎𝐴) → 𝑎 ≤ sup(𝐴, ℝ, < ))
9190adantrr 749 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → 𝑎 ≤ sup(𝐴, ℝ, < ))
924, 5, 63jca 1235 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥))
93 suprub 10863 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐵 ⊆ ℝ ∧ 𝐵 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐵 𝑦𝑥) ∧ 𝑏𝐵) → 𝑏 ≤ sup(𝐵, ℝ, < ))
9492, 93sylan 487 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑏𝐵) → 𝑏 ≤ sup(𝐵, ℝ, < ))
9594adantrl 748 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → 𝑏 ≤ sup(𝐵, ℝ, < ))
9684, 85, 86, 87, 91, 95le2addd 10525 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → (𝑎 + 𝑏) ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )))
9796ex 449 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑎𝐴𝑏𝐵) → (𝑎 + 𝑏) ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < ))))
98 breq1 4586 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝑎 + 𝑏) → (𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) ↔ (𝑎 + 𝑏) ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < ))))
9998biimprcd 239 . . . . . . . . . . . . . . . . . . 19 ((𝑎 + 𝑏) ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) → (𝑤 = (𝑎 + 𝑏) → 𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < ))))
10097, 99syl6 34 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑎𝐴𝑏𝐵) → (𝑤 = (𝑎 + 𝑏) → 𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )))))
101100rexlimdvv 3019 . . . . . . . . . . . . . . . . 17 (𝜑 → (∃𝑎𝐴𝑏𝐵 𝑤 = (𝑎 + 𝑏) → 𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < ))))
10252, 101syl5bi 231 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑤𝐶𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < ))))
103102ralrimiv 2948 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑤𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )))
104 breq2 4587 . . . . . . . . . . . . . . . . 17 (𝑥 = (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) → (𝑤𝑥𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < ))))
105104ralbidv 2969 . . . . . . . . . . . . . . . 16 (𝑥 = (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) → (∀𝑤𝐶 𝑤𝑥 ↔ ∀𝑤𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < ))))
106105rspcev 3282 . . . . . . . . . . . . . . 15 (((sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) ∈ ℝ ∧ ∀𝑤𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < ))) → ∃𝑥 ∈ ℝ ∀𝑤𝐶 𝑤𝑥)
10783, 103, 106syl2anc 691 . . . . . . . . . . . . . 14 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤𝐶 𝑤𝑥)
108 suprub 10863 . . . . . . . . . . . . . . 15 (((𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝐶 𝑤𝑥) ∧ 𝑤𝐶) → 𝑤 ≤ sup(𝐶, ℝ, < ))
109108ex 449 . . . . . . . . . . . . . 14 ((𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝐶 𝑤𝑥) → (𝑤𝐶𝑤 ≤ sup(𝐶, ℝ, < )))
11064, 80, 107, 109syl3anc 1318 . . . . . . . . . . . . 13 (𝜑 → (𝑤𝐶𝑤 ≤ sup(𝐶, ℝ, < )))
11154, 110sylan9r 688 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → (∃𝑏𝐵 𝑤 = (𝑎 + 𝑏) → 𝑤 ≤ sup(𝐶, ℝ, < )))
11243, 111syl5bi 231 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → (𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)} → 𝑤 ≤ sup(𝐶, ℝ, < )))
113112ralrimiv 2948 . . . . . . . . . 10 ((𝜑𝑎𝐴) → ∀𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}𝑤 ≤ sup(𝐶, ℝ, < ))
11433, 31readdcld 9948 . . . . . . . . . . . . . 14 (((𝜑𝑎𝐴) ∧ 𝑏𝐵) → (𝑎 + 𝑏) ∈ ℝ)
115 eleq1a 2683 . . . . . . . . . . . . . 14 ((𝑎 + 𝑏) ∈ ℝ → (𝑧 = (𝑎 + 𝑏) → 𝑧 ∈ ℝ))
116114, 115syl 17 . . . . . . . . . . . . 13 (((𝜑𝑎𝐴) ∧ 𝑏𝐵) → (𝑧 = (𝑎 + 𝑏) → 𝑧 ∈ ℝ))
117116rexlimdva 3013 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → (∃𝑏𝐵 𝑧 = (𝑎 + 𝑏) → 𝑧 ∈ ℝ))
118117abssdv 3639 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)} ⊆ ℝ)
11965isseti 3182 . . . . . . . . . . . . . . . 16 𝑧 𝑧 = (𝑎 + 𝑏)
120119rgenw 2908 . . . . . . . . . . . . . . 15 𝑏𝐵𝑧 𝑧 = (𝑎 + 𝑏)
121 r19.2z 4012 . . . . . . . . . . . . . . 15 ((𝐵 ≠ ∅ ∧ ∀𝑏𝐵𝑧 𝑧 = (𝑎 + 𝑏)) → ∃𝑏𝐵𝑧 𝑧 = (𝑎 + 𝑏))
1225, 120, 121sylancl 693 . . . . . . . . . . . . . 14 (𝜑 → ∃𝑏𝐵𝑧 𝑧 = (𝑎 + 𝑏))
123 rexcom4 3198 . . . . . . . . . . . . . 14 (∃𝑏𝐵𝑧 𝑧 = (𝑎 + 𝑏) ↔ ∃𝑧𝑏𝐵 𝑧 = (𝑎 + 𝑏))
124122, 123sylib 207 . . . . . . . . . . . . 13 (𝜑 → ∃𝑧𝑏𝐵 𝑧 = (𝑎 + 𝑏))
125 abn0 3908 . . . . . . . . . . . . 13 ({𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)} ≠ ∅ ↔ ∃𝑧𝑏𝐵 𝑧 = (𝑎 + 𝑏))
126124, 125sylibr 223 . . . . . . . . . . . 12 (𝜑 → {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)} ≠ ∅)
127126adantr 480 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)} ≠ ∅)
128 suprcl 10862 . . . . . . . . . . . . . 14 ((𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝐶 𝑤𝑥) → sup(𝐶, ℝ, < ) ∈ ℝ)
12964, 80, 107, 128syl3anc 1318 . . . . . . . . . . . . 13 (𝜑 → sup(𝐶, ℝ, < ) ∈ ℝ)
130129adantr 480 . . . . . . . . . . . 12 ((𝜑𝑎𝐴) → sup(𝐶, ℝ, < ) ∈ ℝ)
131 breq2 4587 . . . . . . . . . . . . . 14 (𝑥 = sup(𝐶, ℝ, < ) → (𝑤𝑥𝑤 ≤ sup(𝐶, ℝ, < )))
132131ralbidv 2969 . . . . . . . . . . . . 13 (𝑥 = sup(𝐶, ℝ, < ) → (∀𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}𝑤𝑥 ↔ ∀𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}𝑤 ≤ sup(𝐶, ℝ, < )))
133132rspcev 3282 . . . . . . . . . . . 12 ((sup(𝐶, ℝ, < ) ∈ ℝ ∧ ∀𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}𝑤 ≤ sup(𝐶, ℝ, < )) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}𝑤𝑥)
134130, 113, 133syl2anc 691 . . . . . . . . . . 11 ((𝜑𝑎𝐴) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}𝑤𝑥)
135 suprleub 10866 . . . . . . . . . . 11 ((({𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)} ⊆ ℝ ∧ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}𝑤𝑥) ∧ sup(𝐶, ℝ, < ) ∈ ℝ) → (sup({𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}, ℝ, < ) ≤ sup(𝐶, ℝ, < ) ↔ ∀𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}𝑤 ≤ sup(𝐶, ℝ, < )))
136118, 127, 134, 130, 135syl31anc 1321 . . . . . . . . . 10 ((𝜑𝑎𝐴) → (sup({𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}, ℝ, < ) ≤ sup(𝐶, ℝ, < ) ↔ ∀𝑤 ∈ {𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}𝑤 ≤ sup(𝐶, ℝ, < )))
137113, 136mpbird 246 . . . . . . . . 9 ((𝜑𝑎𝐴) → sup({𝑧 ∣ ∃𝑏𝐵 𝑧 = (𝑎 + 𝑏)}, ℝ, < ) ≤ sup(𝐶, ℝ, < ))
13840, 137eqbrtrd 4605 . . . . . . . 8 ((𝜑𝑎𝐴) → (sup(𝐵, ℝ, < ) + 𝑎) ≤ sup(𝐶, ℝ, < ))
139 breq1 4586 . . . . . . . 8 (𝑤 = (sup(𝐵, ℝ, < ) + 𝑎) → (𝑤 ≤ sup(𝐶, ℝ, < ) ↔ (sup(𝐵, ℝ, < ) + 𝑎) ≤ sup(𝐶, ℝ, < )))
140138, 139syl5ibrcom 236 . . . . . . 7 ((𝜑𝑎𝐴) → (𝑤 = (sup(𝐵, ℝ, < ) + 𝑎) → 𝑤 ≤ sup(𝐶, ℝ, < )))
141140rexlimdva 3013 . . . . . 6 (𝜑 → (∃𝑎𝐴 𝑤 = (sup(𝐵, ℝ, < ) + 𝑎) → 𝑤 ≤ sup(𝐶, ℝ, < )))
14224, 141syl5bi 231 . . . . 5 (𝜑 → (𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)} → 𝑤 ≤ sup(𝐶, ℝ, < )))
143142ralrimiv 2948 . . . 4 (𝜑 → ∀𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}𝑤 ≤ sup(𝐶, ℝ, < ))
14413, 11readdcld 9948 . . . . . . . 8 ((𝜑𝑎𝐴) → (sup(𝐵, ℝ, < ) + 𝑎) ∈ ℝ)
145 eleq1a 2683 . . . . . . . 8 ((sup(𝐵, ℝ, < ) + 𝑎) ∈ ℝ → (𝑧 = (sup(𝐵, ℝ, < ) + 𝑎) → 𝑧 ∈ ℝ))
146144, 145syl 17 . . . . . . 7 ((𝜑𝑎𝐴) → (𝑧 = (sup(𝐵, ℝ, < ) + 𝑎) → 𝑧 ∈ ℝ))
147146rexlimdva 3013 . . . . . 6 (𝜑 → (∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎) → 𝑧 ∈ ℝ))
148147abssdv 3639 . . . . 5 (𝜑 → {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)} ⊆ ℝ)
149 ovex 6577 . . . . . . . . . 10 (sup(𝐵, ℝ, < ) + 𝑎) ∈ V
150149isseti 3182 . . . . . . . . 9 𝑧 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)
151150rgenw 2908 . . . . . . . 8 𝑎𝐴𝑧 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)
152 r19.2z 4012 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ ∀𝑎𝐴𝑧 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)) → ∃𝑎𝐴𝑧 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎))
1532, 151, 152sylancl 693 . . . . . . 7 (𝜑 → ∃𝑎𝐴𝑧 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎))
154 rexcom4 3198 . . . . . . 7 (∃𝑎𝐴𝑧 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎) ↔ ∃𝑧𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎))
155153, 154sylib 207 . . . . . 6 (𝜑 → ∃𝑧𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎))
156 abn0 3908 . . . . . 6 ({𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)} ≠ ∅ ↔ ∃𝑧𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎))
157155, 156sylibr 223 . . . . 5 (𝜑 → {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)} ≠ ∅)
158131ralbidv 2969 . . . . . . 7 (𝑥 = sup(𝐶, ℝ, < ) → (∀𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}𝑤𝑥 ↔ ∀𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}𝑤 ≤ sup(𝐶, ℝ, < )))
159158rspcev 3282 . . . . . 6 ((sup(𝐶, ℝ, < ) ∈ ℝ ∧ ∀𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}𝑤 ≤ sup(𝐶, ℝ, < )) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}𝑤𝑥)
160129, 143, 159syl2anc 691 . . . . 5 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}𝑤𝑥)
161 suprleub 10866 . . . . 5 ((({𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)} ⊆ ℝ ∧ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}𝑤𝑥) ∧ sup(𝐶, ℝ, < ) ∈ ℝ) → (sup({𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}, ℝ, < ) ≤ sup(𝐶, ℝ, < ) ↔ ∀𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}𝑤 ≤ sup(𝐶, ℝ, < )))
162148, 157, 160, 129, 161syl31anc 1321 . . . 4 (𝜑 → (sup({𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}, ℝ, < ) ≤ sup(𝐶, ℝ, < ) ↔ ∀𝑤 ∈ {𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}𝑤 ≤ sup(𝐶, ℝ, < )))
163143, 162mpbird 246 . . 3 (𝜑 → sup({𝑧 ∣ ∃𝑎𝐴 𝑧 = (sup(𝐵, ℝ, < ) + 𝑎)}, ℝ, < ) ≤ sup(𝐶, ℝ, < ))
16420, 163eqbrtrd 4605 . 2 (𝜑 → (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) ≤ sup(𝐶, ℝ, < ))
165 suprleub 10866 . . . 4 (((𝐶 ⊆ ℝ ∧ 𝐶 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤𝐶 𝑤𝑥) ∧ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) ∈ ℝ) → (sup(𝐶, ℝ, < ) ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) ↔ ∀𝑤𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < ))))
16664, 80, 107, 83, 165syl31anc 1321 . . 3 (𝜑 → (sup(𝐶, ℝ, < ) ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) ↔ ∀𝑤𝐶 𝑤 ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < ))))
167103, 166mpbird 246 . 2 (𝜑 → sup(𝐶, ℝ, < ) ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )))
16883, 129letri3d 10058 . 2 (𝜑 → ((sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < ) ↔ ((sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) ≤ sup(𝐶, ℝ, < ) ∧ sup(𝐶, ℝ, < ) ≤ (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )))))
169164, 167, 168mpbir2and 959 1 (𝜑 → (sup(𝐴, ℝ, < ) + sup(𝐵, ℝ, < )) = sup(𝐶, ℝ, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  (class class class)co 6549  supcsup 8229  ℝcr 9814   + caddc 9818   < clt 9953   ≤ cle 9954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148 This theorem is referenced by:  ismblfin  32620  itg2addnc  32634  sge0resplit  39299
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