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Theorem inelcarsg 30347
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 30341 . . . . . 6 (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))))
51, 4mpbid 222 . . . . 5 (𝜑 → (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)))
65simpld 475 . . . 4 (𝜑𝐴𝑂)
7 ssinss1 3833 . . . 4 (𝐴𝑂 → (𝐴𝐵) ⊆ 𝑂)
86, 7syl 17 . . 3 (𝜑 → (𝐴𝐵) ⊆ 𝑂)
9 iccssxr 12241 . . . . . . . . 9 (0[,]+∞) ⊆ ℝ*
103adantr 481 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
11 simpr 477 . . . . . . . . . . 11 ((𝜑𝑒 ∈ 𝒫 𝑂) → 𝑒 ∈ 𝒫 𝑂)
1211elpwdifcl 29330 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑒 ∖ (𝐴𝐵)) ∈ 𝒫 𝑂)
1310, 12ffvelrnd 6346 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ (𝐴𝐵))) ∈ (0[,]+∞))
149, 13sseldi 3593 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ (𝐴𝐵))) ∈ ℝ*)
1511elpwincl1 29329 . . . . . . . . . . . 12 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑒𝐴) ∈ 𝒫 𝑂)
1615elpwdifcl 29330 . . . . . . . . . . 11 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑒𝐴) ∖ 𝐵) ∈ 𝒫 𝑂)
1710, 16ffvelrnd 6346 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘((𝑒𝐴) ∖ 𝐵)) ∈ (0[,]+∞))
189, 17sseldi 3593 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘((𝑒𝐴) ∖ 𝐵)) ∈ ℝ*)
1911elpwdifcl 29330 . . . . . . . . . . 11 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑒𝐴) ∈ 𝒫 𝑂)
2010, 19ffvelrnd 6346 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒𝐴)) ∈ (0[,]+∞))
219, 20sseldi 3593 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒𝐴)) ∈ ℝ*)
2218, 21xaddcld 12116 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))) ∈ ℝ*)
2311elpwincl1 29329 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑒 ∩ (𝐴𝐵)) ∈ 𝒫 𝑂)
2410, 23ffvelrnd 6346 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∩ (𝐴𝐵))) ∈ (0[,]+∞))
259, 24sseldi 3593 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∩ (𝐴𝐵))) ∈ ℝ*)
26 indifundif 29328 . . . . . . . . . 10 (((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴)) = (𝑒 ∖ (𝐴𝐵))
2726fveq2i 6181 . . . . . . . . 9 (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴))) = (𝑀‘(𝑒 ∖ (𝐴𝐵)))
28 inelcarsg.1 . . . . . . . . . . . . 13 ((𝜑𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))
29283expb 1264 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂)) → (𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))
3029ralrimivva 2968 . . . . . . . . . . 11 (𝜑 → ∀𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂(𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))
3130adantr 481 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → ∀𝑎 ∈ 𝒫 𝑂𝑏 ∈ 𝒫 𝑂(𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)))
32 uneq1 3752 . . . . . . . . . . . . . 14 (𝑎 = ((𝑒𝐴) ∖ 𝐵) → (𝑎𝑏) = (((𝑒𝐴) ∖ 𝐵) ∪ 𝑏))
3332fveq2d 6182 . . . . . . . . . . . . 13 (𝑎 = ((𝑒𝐴) ∖ 𝐵) → (𝑀‘(𝑎𝑏)) = (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ 𝑏)))
34 fveq2 6178 . . . . . . . . . . . . . 14 (𝑎 = ((𝑒𝐴) ∖ 𝐵) → (𝑀𝑎) = (𝑀‘((𝑒𝐴) ∖ 𝐵)))
3534oveq1d 6650 . . . . . . . . . . . . 13 (𝑎 = ((𝑒𝐴) ∖ 𝐵) → ((𝑀𝑎) +𝑒 (𝑀𝑏)) = ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀𝑏)))
3633, 35breq12d 4657 . . . . . . . . . . . 12 (𝑎 = ((𝑒𝐴) ∖ 𝐵) → ((𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)) ↔ (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ 𝑏)) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀𝑏))))
37 uneq2 3753 . . . . . . . . . . . . . 14 (𝑏 = (𝑒𝐴) → (((𝑒𝐴) ∖ 𝐵) ∪ 𝑏) = (((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴)))
3837fveq2d 6182 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝐴) → (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ 𝑏)) = (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴))))
39 fveq2 6178 . . . . . . . . . . . . . 14 (𝑏 = (𝑒𝐴) → (𝑀𝑏) = (𝑀‘(𝑒𝐴)))
4039oveq2d 6651 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝐴) → ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀𝑏)) = ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))))
4138, 40breq12d 4657 . . . . . . . . . . . 12 (𝑏 = (𝑒𝐴) → ((𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ 𝑏)) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀𝑏)) ↔ (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴))) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))))
4236, 41rspc2v 3317 . . . . . . . . . . 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 445 . . . . . . . . . 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 1323 . . . . . . . . 9 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(((𝑒𝐴) ∖ 𝐵) ∪ (𝑒𝐴))) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))))
4527, 44syl5eqbrr 4680 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∖ (𝐴𝐵))) ≤ ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))))
46 xleadd2a 12069 . . . . . . . 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 1327 . . . . . . 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 30341 . . . . . . . . . . . . 13 (𝜑 → (𝐵 ∈ (toCaraSiga‘𝑀) ↔ (𝐵𝑂 ∧ ∀𝑓 ∈ 𝒫 𝑂((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓))))
5048, 49mpbid 222 . . . . . . . . . . . 12 (𝜑 → (𝐵𝑂 ∧ ∀𝑓 ∈ 𝒫 𝑂((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓)))
5150simprd 479 . . . . . . . . . . 11 (𝜑 → ∀𝑓 ∈ 𝒫 𝑂((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓))
5251adantr 481 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → ∀𝑓 ∈ 𝒫 𝑂((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓))
53 ineq1 3799 . . . . . . . . . . . . . . 15 (𝑓 = (𝑒𝐴) → (𝑓𝐵) = ((𝑒𝐴) ∩ 𝐵))
5453fveq2d 6182 . . . . . . . . . . . . . 14 (𝑓 = (𝑒𝐴) → (𝑀‘(𝑓𝐵)) = (𝑀‘((𝑒𝐴) ∩ 𝐵)))
55 difeq1 3713 . . . . . . . . . . . . . . 15 (𝑓 = (𝑒𝐴) → (𝑓𝐵) = ((𝑒𝐴) ∖ 𝐵))
5655fveq2d 6182 . . . . . . . . . . . . . 14 (𝑓 = (𝑒𝐴) → (𝑀‘(𝑓𝐵)) = (𝑀‘((𝑒𝐴) ∖ 𝐵)))
5754, 56oveq12d 6653 . . . . . . . . . . . . 13 (𝑓 = (𝑒𝐴) → ((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))))
58 fveq2 6178 . . . . . . . . . . . . 13 (𝑓 = (𝑒𝐴) → (𝑀𝑓) = (𝑀‘(𝑒𝐴)))
5957, 58eqeq12d 2635 . . . . . . . . . . . 12 (𝑓 = (𝑒𝐴) → (((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓) ↔ ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) = (𝑀‘(𝑒𝐴))))
6059adantl 482 . . . . . . . . . . 11 (((𝜑𝑒 ∈ 𝒫 𝑂) ∧ 𝑓 = (𝑒𝐴)) → (((𝑀‘(𝑓𝐵)) +𝑒 (𝑀‘(𝑓𝐵))) = (𝑀𝑓) ↔ ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) = (𝑀‘(𝑒𝐴))))
6115, 60rspcdv 3307 . . . . . . . . . 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 6650 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → (((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 (𝑀‘((𝑒𝐴) ∖ 𝐵))) +𝑒 (𝑀‘(𝑒𝐴))) = ((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))))
6415elpwincl1 29329 . . . . . . . . . . 11 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑒𝐴) ∩ 𝐵) ∈ 𝒫 𝑂)
6510, 64ffvelrnd 6346 . . . . . . . . . 10 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘((𝑒𝐴) ∩ 𝐵)) ∈ (0[,]+∞))
66 xrge0addass 29664 . . . . . . . . . 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 1324 . . . . . . . . 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 3815 . . . . . . . . . . 11 ((𝑒𝐴) ∩ 𝐵) = (𝑒 ∩ (𝐴𝐵))
6968fveq2i 6181 . . . . . . . . . 10 (𝑀‘((𝑒𝐴) ∩ 𝐵)) = (𝑀‘(𝑒 ∩ (𝐴𝐵)))
7069oveq1i 6645 . . . . . . . . 9 ((𝑀‘((𝑒𝐴) ∩ 𝐵)) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))) = ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴))))
7167, 70syl6eq 2670 . . . . . . . 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 479 . . . . . . . . 9 (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))
7372r19.21bi 2929 . . . . . . . 8 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))
7463, 71, 733eqtr3d 2662 . . . . . . 7 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 ((𝑀‘((𝑒𝐴) ∖ 𝐵)) +𝑒 (𝑀‘(𝑒𝐴)))) = (𝑀𝑒))
7547, 74breqtrd 4670 . . . . . 6 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ≤ (𝑀𝑒))
76 inundif 4037 . . . . . . . 8 ((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵))) = 𝑒
7776fveq2i 6181 . . . . . . 7 (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒)
78 uneq1 3752 . . . . . . . . . . . 12 (𝑎 = (𝑒 ∩ (𝐴𝐵)) → (𝑎𝑏) = ((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏))
7978fveq2d 6182 . . . . . . . . . . 11 (𝑎 = (𝑒 ∩ (𝐴𝐵)) → (𝑀‘(𝑎𝑏)) = (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏)))
80 fveq2 6178 . . . . . . . . . . . 12 (𝑎 = (𝑒 ∩ (𝐴𝐵)) → (𝑀𝑎) = (𝑀‘(𝑒 ∩ (𝐴𝐵))))
8180oveq1d 6650 . . . . . . . . . . 11 (𝑎 = (𝑒 ∩ (𝐴𝐵)) → ((𝑀𝑎) +𝑒 (𝑀𝑏)) = ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀𝑏)))
8279, 81breq12d 4657 . . . . . . . . . 10 (𝑎 = (𝑒 ∩ (𝐴𝐵)) → ((𝑀‘(𝑎𝑏)) ≤ ((𝑀𝑎) +𝑒 (𝑀𝑏)) ↔ (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏)) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀𝑏))))
83 uneq2 3753 . . . . . . . . . . . 12 (𝑏 = (𝑒 ∖ (𝐴𝐵)) → ((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏) = ((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵))))
8483fveq2d 6182 . . . . . . . . . . 11 (𝑏 = (𝑒 ∖ (𝐴𝐵)) → (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ 𝑏)) = (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵)))))
85 fveq2 6178 . . . . . . . . . . . 12 (𝑏 = (𝑒 ∖ (𝐴𝐵)) → (𝑀𝑏) = (𝑀‘(𝑒 ∖ (𝐴𝐵))))
8685oveq2d 6651 . . . . . . . . . . 11 (𝑏 = (𝑒 ∖ (𝐴𝐵)) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀𝑏)) = ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))))
8784, 86breq12d 4657 . . . . . . . . . 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 3317 . . . . . . . . 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 445 . . . . . . . 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 1323 . . . . . . 7 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀‘((𝑒 ∩ (𝐴𝐵)) ∪ (𝑒 ∖ (𝐴𝐵)))) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))))
9177, 90syl5eqbrr 4680 . . . . . 6 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀𝑒) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))))
9275, 91jca 554 . . . . 5 ((𝜑𝑒 ∈ 𝒫 𝑂) → (((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ≤ (𝑀𝑒) ∧ (𝑀𝑒) ≤ ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵))))))
9325, 14xaddcld 12116 . . . . . 6 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) ∈ ℝ*)
943ffvelrnda 6345 . . . . . . 7 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀𝑒) ∈ (0[,]+∞))
959, 94sseldi 3593 . . . . . 6 ((𝜑𝑒 ∈ 𝒫 𝑂) → (𝑀𝑒) ∈ ℝ*)
96 xrletri3 11970 . . . . . 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 692 . . . . 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 247 . . . 4 ((𝜑𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒))
9998ralrimiva 2963 . . 3 (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒))
1008, 99jca 554 . 2 (𝜑 → ((𝐴𝐵) ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒)))
1012, 3elcarsg 30341 . 2 (𝜑 → ((𝐴𝐵) ∈ (toCaraSiga‘𝑀) ↔ ((𝐴𝐵) ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ (𝐴𝐵))) +𝑒 (𝑀‘(𝑒 ∖ (𝐴𝐵)))) = (𝑀𝑒))))
102100, 101mpbird 247 1 (𝜑 → (𝐴𝐵) ∈ (toCaraSiga‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  cdif 3564  cun 3565  cin 3566  wss 3567  𝒫 cpw 4149   class class class wbr 4644  wf 5872  cfv 5876  (class class class)co 6635  0cc0 9921  +∞cpnf 10056  *cxr 10058  cle 10060   +𝑒 cxad 11929  [,]cicc 12163  toCaraSigaccarsg 30337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-po 5025  df-so 5026  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-xadd 11932  df-icc 12167  df-carsg 30338
This theorem is referenced by:  unelcarsg  30348  difelcarsg2  30349
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