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Theorem inelcarsg 29503
Description: The Caratheodory measurable sets are closed under intersection. (Contributed by Thierry Arnoux, 18-May-2020.)
Hypotheses
Ref Expression
carsgval.1 (𝜑𝑂𝑉)
carsgval.2 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
difelcarsg.1 (𝜑𝐴 ∈ (toCaraSiga‘𝑀))
inelcarsg.1 ((𝜑𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))
inelcarsg.2 (𝜑𝐵 ∈ (toCaraSiga‘𝑀))
Assertion
Ref Expression
inelcarsg (𝜑 → (𝐴𝐵) ∈ (toCaraSiga‘𝑀))
Distinct variable groups:   𝑀,𝑎   𝑂,𝑎   𝜑,𝑎   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏   𝑀,𝑏   𝑂,𝑏   𝜑,𝑏
Allowed substitution hints:   𝑉(𝑎,𝑏)

Proof of Theorem inelcarsg
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difelcarsg.1 . . . . . 6 (𝜑𝐴 ∈ (toCaraSiga‘𝑀))
2 carsgval.1 . . . . . . 7 (𝜑𝑂𝑉)
3 carsgval.2 . . . . . . 7 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
42, 3elcarsg 29497 . . . . . 6 (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))))
51, 4mpbid 220 . . . . 5 (𝜑 → (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)))
65simpld 473 . . . 4 (𝜑𝐴𝑂)
7 ssinss1 3799 . . . 4 (𝐴𝑂 → (𝐴𝐵) ⊆ 𝑂)
86, 7syl 17 . . 3 (𝜑 → (𝐴𝐵) ⊆ 𝑂)
9 iccssxr 12080 . . . . . . . . 9 (0[,]+∞) ⊆ ℝ*
103adantr 479 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
11 simpr 475 . . . . . . . . . . 11 ((𝜑𝑒 ∈ 𝒫 𝑂) → 𝑒 ∈ 𝒫 𝑂)
1211elpwdifcl 28545 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑒 ∖ (𝐴𝐵)) ∈ 𝒫 𝑂)
1310, 12ffvelrnd 6250 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ (𝐴𝐵))) ∈ (0[,]+∞))
149, 13sseldi 3562 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ (𝐴𝐵))) ∈ ℝ*)
1511elpwincl1 28544 . . . . . . . . . . . 12 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑒𝐴) ∈ 𝒫 𝑂)
1615elpwdifcl 28545 . . . . . . . . . . 11 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑒𝐴) ∖ 𝐵) ∈ 𝒫 𝑂)
1710, 16ffvelrnd 6250 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘((𝑒𝐴) ∖ 𝐵)) ∈ (0[,]+∞))
189, 17sseldi 3562 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘((𝑒𝐴) ∖ 𝐵)) ∈ ℝ*)
1911elpwdifcl 28545 . . . . . . . . . . 11 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑒𝐴) ∈ 𝒫 𝑂)
2010, 19ffvelrnd 6250 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒𝐴)) ∈ (0[,]+∞))
219, 20sseldi 3562 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒𝐴)) ∈ ℝ*)
2218, 21xaddcld 11957 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))) ∈ ℝ*)
2311elpwincl1 28544 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑒 ∩ (𝐴𝐵)) ∈ 𝒫 𝑂)
2410, 23ffvelrnd 6250 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∩ (𝐴𝐵))) ∈ (0[,]+∞))
259, 24sseldi 3562 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∩ (𝐴𝐵))) ∈ ℝ*)
26 indifundif 28543 . . . . . . . . . 10 (((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴)) = (𝑒 ∖ (𝐴𝐵))
2726fveq2i 6088 . . . . . . . . 9 (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴))) = (𝑀‘(𝑒 ∖ (𝐴𝐵)))
28 inelcarsg.1 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))
29283expb 1257 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂)) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))
3029ralrimivva 2950 . . . . . . . . . . 11 (𝜑 → ∀𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂(𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))
3130adantr 479 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → ∀𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂(𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))
32 uneq1 3718 . . . . . . . . . . . . . 14 (𝑎 = ((𝑒𝐴) ∖ 𝐵) → (𝑎𝑏) = (((𝑒𝐴) ∖ 𝐵) ∪ 𝑏))
3332fveq2d 6089 . . . . . . . . . . . . 13 (𝑎 = ((𝑒𝐴) ∖ 𝐵) → (𝑀‘(𝑎𝑏)) = (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ 𝑏)))
34 fveq2 6085 . . . . . . . . . . . . . 14 (𝑎 = ((𝑒𝐴) ∖ 𝐵) → (𝑀𝑎) = (𝑀‘((𝑒𝐴) ∖ 𝐵)))
3534oveq1d 6539 . . . . . . . . . . . . 13 (𝑎 = ((𝑒𝐴) ∖ 𝐵) → ((𝑀𝑎) +𝑒 (𝑀𝑏)) = ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀𝑏)))
3633, 35breq12d 4587 . . . . . . . . . . . 12 (𝑎 = ((𝑒𝐴) ∖ 𝐵) → ((𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)) ↔ (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ 𝑏)) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀𝑏))))
37 uneq2 3719 . . . . . . . . . . . . . 14 (𝑏 = (𝑒𝐴) → (((𝑒𝐴) ∖ 𝐵) ∪ 𝑏) = (((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴)))
3837fveq2d 6089 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝐴) → (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ 𝑏)) = (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴))))
39 fveq2 6085 . . . . . . . . . . . . . 14 (𝑏 = (𝑒𝐴) → (𝑀𝑏) = (𝑀‘(𝑒𝐴)))
4039oveq2d 6540 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝐴) → ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀𝑏)) = ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))))
4138, 40breq12d 4587 . . . . . . . . . . . 12 (𝑏 = (𝑒𝐴) → ((𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ 𝑏)) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀𝑏)) ↔ (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴))) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))))
4236, 41rspc2v 3289 . . . . . . . . . . 11 ((((𝑒𝐴) ∖ 𝐵) ∈ 𝒫 𝑂 ∧ (𝑒𝐴) ∈ 𝒫 𝑂) → (∀𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂(𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)) → (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴))) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))))
4342imp 443 . . . . . . . . . 10 (((((𝑒𝐴) ∖ 𝐵) ∈ 𝒫 𝑂 ∧ (𝑒𝐴) ∈ 𝒫 𝑂) ∧ ∀𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂(𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏))) → (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴))) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))))
4416, 19, 31, 43syl21anc 1316 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴))) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))))
4527, 44syl5eqbrr 4610 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ (𝐴𝐵))) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))))
46 xleadd2a 11910 . . . . . . . 8 ((((𝑀‘(𝑒 ∖ (𝐴𝐵))) ∈ ℝ* ∧ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))) ∈ ℝ* ∧ (𝑀‘(𝑒 ∩ (𝐴𝐵))) ∈ ℝ*) ∧ (𝑀‘(𝑒 ∖ (𝐴𝐵))) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))))
4714, 22, 25, 45, 46syl31anc 1320 . . . . . . 7 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))))
48 inelcarsg.2 . . . . . . . . . . . . 13 (𝜑𝐵 ∈ (toCaraSiga‘𝑀))
492, 3elcarsg 29497 . . . . . . . . . . . . 13 (𝜑 → (𝐵 ∈ (toCaraSiga‘𝑀) ↔ (𝐵𝑂 ∧ ∀𝑓 ∈ 𝒫 𝑂((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓))))
5048, 49mpbid 220 . . . . . . . . . . . 12 (𝜑 → (𝐵𝑂 ∧ ∀𝑓 ∈ 𝒫 𝑂((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓)))
5150simprd 477 . . . . . . . . . . 11 (𝜑 → ∀𝑓 ∈ 𝒫 𝑂((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓))
5251adantr 479 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → ∀𝑓 ∈ 𝒫 𝑂((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓))
53 ineq1 3765 . . . . . . . . . . . . . . 15 (𝑓 = (𝑒𝐴) → (𝑓𝐵) = ((𝑒𝐴) ∩ 𝐵))
5453fveq2d 6089 . . . . . . . . . . . . . 14 (𝑓 = (𝑒𝐴) → (𝑀‘(𝑓𝐵)) = (𝑀‘((𝑒𝐴) ∩ 𝐵)))
55 difeq1 3679 . . . . . . . . . . . . . . 15 (𝑓 = (𝑒𝐴) → (𝑓𝐵) = ((𝑒𝐴) ∖ 𝐵))
5655fveq2d 6089 . . . . . . . . . . . . . 14 (𝑓 = (𝑒𝐴) → (𝑀‘(𝑓𝐵)) = (𝑀‘((𝑒𝐴) ∖ 𝐵)))
5754, 56oveq12d 6542 . . . . . . . . . . . . 13 (𝑓 = (𝑒𝐴) → ((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))))
58 fveq2 6085 . . . . . . . . . . . . 13 (𝑓 = (𝑒𝐴) → (𝑀𝑓) = (𝑀‘(𝑒𝐴)))
5957, 58eqeq12d 2621 . . . . . . . . . . . 12 (𝑓 = (𝑒𝐴) → (((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓) ↔ ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) = (𝑀‘(𝑒𝐴))))
6059adantl 480 . . . . . . . . . . 11 (((𝜑𝑒 ∈ 𝒫 𝑂) ∧ 𝑓 = (𝑒𝐴)) → (((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓) ↔ ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) = (𝑀‘(𝑒𝐴))))
6115, 60rspcdv 3281 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (∀𝑓 ∈ 𝒫 𝑂((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓) → ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) = (𝑀‘(𝑒𝐴))))
6252, 61mpd 15 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) = (𝑀‘(𝑒𝐴)))
6362oveq1d 6539 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → (((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) +𝑒 (𝑀‘(𝑒𝐴))) = ((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))))
6415elpwincl1 28544 . . . . . . . . . . 11 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑒𝐴) ∩ 𝐵) ∈ 𝒫 𝑂)
6510, 64ffvelrnd 6250 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘((𝑒𝐴) ∩ 𝐵)) ∈ (0[,]+∞))
66 xrge0addass 28824 . . . . . . . . . 10 (((𝑀‘((𝑒𝐴) ∩ 𝐵)) ∈ (0[,]+∞) ∧ (𝑀‘((𝑒𝐴) ∖ 𝐵)) ∈ (0[,]+∞) ∧ (𝑀‘(𝑒𝐴)) ∈ (0[,]+∞)) → (((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) +𝑒 (𝑀‘(𝑒𝐴))) = ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))))
6765, 17, 20, 66syl3anc 1317 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) +𝑒 (𝑀‘(𝑒𝐴))) = ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))))
68 inass 3781 . . . . . . . . . . 11 ((𝑒𝐴) ∩ 𝐵) = (𝑒 ∩ (𝐴𝐵))
6968fveq2i 6088 . . . . . . . . . 10 (𝑀‘((𝑒𝐴) ∩ 𝐵)) = (𝑀‘(𝑒 ∩ (𝐴𝐵)))
7069oveq1i 6534 . . . . . . . . 9 ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))) = ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))))
7167, 70syl6eq 2656 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → (((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) +𝑒 (𝑀‘(𝑒𝐴))) = ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))))
725simprd 477 . . . . . . . . 9 (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))
7372r19.21bi 2912 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))
7463, 71, 733eqtr3d 2648 . . . . . . 7 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))) = (𝑀𝑒))
7547, 74breqtrd 4600 . . . . . 6 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ≤ (𝑀𝑒))
76 inundif 3994 . . . . . . . 8 ((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵))) = 𝑒
7776fveq2i 6088 . . . . . . 7 (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒)
78 uneq1 3718 . . . . . . . . . . . 12 (𝑎 = (𝑒 ∩ (𝐴𝐵)) → (𝑎𝑏) = ((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏))
7978fveq2d 6089 . . . . . . . . . . 11 (𝑎 = (𝑒 ∩ (𝐴𝐵)) → (𝑀‘(𝑎𝑏)) = (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏)))
80 fveq2 6085 . . . . . . . . . . . 12 (𝑎 = (𝑒 ∩ (𝐴𝐵)) → (𝑀𝑎) = (𝑀‘(𝑒 ∩ (𝐴𝐵))))
8180oveq1d 6539 . . . . . . . . . . 11 (𝑎 = (𝑒 ∩ (𝐴𝐵)) → ((𝑀𝑎) +𝑒 (𝑀𝑏)) = ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀𝑏)))
8279, 81breq12d 4587 . . . . . . . . . 10 (𝑎 = (𝑒 ∩ (𝐴𝐵)) → ((𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)) ↔ (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏)) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀𝑏))))
83 uneq2 3719 . . . . . . . . . . . 12 (𝑏 = (𝑒 ∖ (𝐴𝐵)) → ((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏) = ((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵))))
8483fveq2d 6089 . . . . . . . . . . 11 (𝑏 = (𝑒 ∖ (𝐴𝐵)) → (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏)) = (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵)))))
85 fveq2 6085 . . . . . . . . . . . 12 (𝑏 = (𝑒 ∖ (𝐴𝐵)) → (𝑀𝑏) = (𝑀‘(𝑒 ∖ (𝐴𝐵))))
8685oveq2d 6540 . . . . . . . . . . 11 (𝑏 = (𝑒 ∖ (𝐴𝐵)) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀𝑏)) = ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))))
8784, 86breq12d 4587 . . . . . . . . . 10 (𝑏 = (𝑒 ∖ (𝐴𝐵)) → ((𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏)) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀𝑏)) ↔ (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵)))) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵))))))
8882, 87rspc2v 3289 . . . . . . . . 9 (((𝑒 ∩ (𝐴𝐵)) ∈ 𝒫 𝑂 ∧ (𝑒 ∖ (𝐴𝐵)) ∈ 𝒫 𝑂) → (∀𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂(𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)) → (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵)))) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵))))))
8988imp 443 . . . . . . . 8 ((((𝑒 ∩ (𝐴𝐵)) ∈ 𝒫 𝑂 ∧ (𝑒 ∖ (𝐴𝐵)) ∈ 𝒫 𝑂) ∧ ∀𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂(𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏))) → (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵)))) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))))
9023, 12, 31, 89syl21anc 1316 . . . . . . 7 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵)))) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))))
9177, 90syl5eqbrr 4610 . . . . . 6 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀𝑒) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))))
9275, 91jca 552 . . . . 5 ((𝜑𝑒 ∈ 𝒫 𝑂) → (((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ≤ (𝑀𝑒) ∧ (𝑀𝑒) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵))))))
9325, 14xaddcld 11957 . . . . . 6 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ∈ ℝ*)
943ffvelrnda 6249 . . . . . . 7 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀𝑒) ∈ (0[,]+∞))
959, 94sseldi 3562 . . . . . 6 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀𝑒) ∈ ℝ*)
96 xrletri3 11817 . . . . . 6 ((((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ∈ ℝ* ∧ (𝑀𝑒) ∈ ℝ*) → (((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒) ↔ (((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ≤ (𝑀𝑒) ∧ (𝑀𝑒) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))))))
9793, 95, 96syl2anc 690 . . . . 5 ((𝜑𝑒 ∈ 𝒫 𝑂) → (((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒) ↔ (((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ≤ (𝑀𝑒) ∧ (𝑀𝑒) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))))))
9892, 97mpbird 245 . . . 4 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒))
9998ralrimiva 2945 . . 3 (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒))
1008, 99jca 552 . 2 (𝜑 → ((𝐴𝐵) ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒)))
1012, 3elcarsg 29497 . 2 (𝜑 → ((𝐴𝐵) ∈ (toCaraSiga‘𝑀) ↔ ((𝐴𝐵) ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒))))
102100, 101mpbird 245 1 (𝜑 → (𝐴𝐵) ∈ (toCaraSiga‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2892  cdif 3533  cun 3534  cin 3535  wss 3536  𝒫 cpw 4104   class class class wbr 4574  wf 5783  cfv 5787  (class class class)co 6524  0cc0 9789  +∞cpnf 9924  *cxr 9926  cle 9928   +𝑒 cxad 11773  [,]cicc 12002  toCaraSigaccarsg 29493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-po 4946  df-so 4947  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-xadd 11776  df-icc 12006  df-carsg 29494
This theorem is referenced by:  unelcarsg  29504  difelcarsg2  29505
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