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Theorem prarloc 6485
Description: A Dedekind cut is arithmetically located. Part of Proposition 11.15 of [BauerTaylor], p. 52, slightly modified. It states that given a tolerance 𝑃, there are elements of the lower and upper cut which are within that tolerance of each other.

Usually, proofs will be shorter if they use prarloc2 6486 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)

Assertion
Ref Expression
prarloc ((⟨𝐿, 𝑈 P 𝑃 Q) → 𝑎 𝐿 𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))
Distinct variable groups:   𝐿,𝑎,𝑏   𝑃,𝑎,𝑏   𝑈,𝑎,𝑏

Proof of Theorem prarloc
Dummy variables 𝑚 𝑛 𝑞 x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prml 6459 . . . . . . 7 (⟨𝐿, 𝑈 Px Q x 𝐿)
2 df-rex 2306 . . . . . . 7 (x Q x 𝐿x(x Q x 𝐿))
31, 2sylib 127 . . . . . 6 (⟨𝐿, 𝑈 Px(x Q x 𝐿))
43adantr 261 . . . . 5 ((⟨𝐿, 𝑈 P 𝑃 Q) → x(x Q x 𝐿))
5 prmu 6460 . . . . . . 7 (⟨𝐿, 𝑈 Py Q y 𝑈)
6 df-rex 2306 . . . . . . 7 (y Q y 𝑈y(y Q y 𝑈))
75, 6sylib 127 . . . . . 6 (⟨𝐿, 𝑈 Py(y Q y 𝑈))
87adantr 261 . . . . 5 ((⟨𝐿, 𝑈 P 𝑃 Q) → y(y Q y 𝑈))
9 subhalfnqq 6397 . . . . . . . . 9 (𝑃 Q𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃)
109adantl 262 . . . . . . . 8 ((⟨𝐿, 𝑈 P 𝑃 Q) → 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃)
11 df-rex 2306 . . . . . . . 8 (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃𝑞(𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))
1210, 11sylib 127 . . . . . . 7 ((⟨𝐿, 𝑈 P 𝑃 Q) → 𝑞(𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))
1312ancli 306 . . . . . 6 ((⟨𝐿, 𝑈 P 𝑃 Q) → ((⟨𝐿, 𝑈 P 𝑃 Q) 𝑞(𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃)))
14 19.42v 1783 . . . . . 6 (𝑞((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃)) ↔ ((⟨𝐿, 𝑈 P 𝑃 Q) 𝑞(𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃)))
1513, 14sylibr 137 . . . . 5 ((⟨𝐿, 𝑈 P 𝑃 Q) → 𝑞((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃)))
16 eeeanv 1805 . . . . 5 (xy𝑞((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) ↔ (x(x Q x 𝐿) y(y Q y 𝑈) 𝑞((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))))
174, 8, 15, 16syl3anbrc 1087 . . . 4 ((⟨𝐿, 𝑈 P 𝑃 Q) → xy𝑞((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))))
18 prarloclemarch2 6402 . . . . . . . . . . . . . 14 ((y Q x Q 𝑞 Q) → 𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))
19 df-rex 2306 . . . . . . . . . . . . . 14 (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))) ↔ 𝑛(𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)))))
2018, 19sylib 127 . . . . . . . . . . . . 13 ((y Q x Q 𝑞 Q) → 𝑛(𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)))))
21203com12 1107 . . . . . . . . . . . 12 ((x Q y Q 𝑞 Q) → 𝑛(𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)))))
22213adant1r 1127 . . . . . . . . . . 11 (((x Q x 𝐿) y Q 𝑞 Q) → 𝑛(𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)))))
23223adant2r 1129 . . . . . . . . . 10 (((x Q x 𝐿) (y Q y 𝑈) 𝑞 Q) → 𝑛(𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)))))
24233adant3r 1131 . . . . . . . . 9 (((x Q x 𝐿) (y Q y 𝑈) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃)) → 𝑛(𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)))))
25243adant3l 1130 . . . . . . . 8 (((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) → 𝑛(𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)))))
2625ancli 306 . . . . . . 7 (((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) → (((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) 𝑛(𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))))
27 19.42v 1783 . . . . . . 7 (𝑛(((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) ↔ (((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) 𝑛(𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))))
2826, 27sylibr 137 . . . . . 6 (((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) → 𝑛(((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))))
29282eximi 1489 . . . . 5 (y𝑞((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) → y𝑞𝑛(((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))))
3029eximi 1488 . . . 4 (xy𝑞((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) → xy𝑞𝑛(((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))))
31 simpl1l 954 . . . . . . . . . 10 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → x Q)
32 simp3rl 976 . . . . . . . . . . 11 (((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) → 𝑞 Q)
3332adantr 261 . . . . . . . . . 10 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → 𝑞 Q)
34 simp3rr 977 . . . . . . . . . . 11 (((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) → (𝑞 +Q 𝑞) <Q 𝑃)
3534adantr 261 . . . . . . . . . 10 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → (𝑞 +Q 𝑞) <Q 𝑃)
3631, 33, 353jca 1083 . . . . . . . . 9 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → (x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))
37 simp3ll 974 . . . . . . . . . . . 12 (((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) → ⟨𝐿, 𝑈 P)
3837adantr 261 . . . . . . . . . . 11 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → ⟨𝐿, 𝑈 P)
39 simpl1r 955 . . . . . . . . . . 11 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → x 𝐿)
40 simprl 483 . . . . . . . . . . 11 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → 𝑛 N)
41 simprrl 491 . . . . . . . . . . 11 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → 1𝑜 <N 𝑛)
42 simprrr 492 . . . . . . . . . . . 12 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)))
43 simpl2r 957 . . . . . . . . . . . . 13 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → y 𝑈)
44 prcunqu 6467 . . . . . . . . . . . . 13 ((⟨𝐿, 𝑈 P y 𝑈) → (y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)) → (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))
4538, 43, 44syl2anc 391 . . . . . . . . . . . 12 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → (y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)) → (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))
4642, 45mpd 13 . . . . . . . . . . 11 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)
47 prarloclem 6483 . . . . . . . . . . 11 (((⟨𝐿, 𝑈 P x 𝐿) (𝑛 N 𝑞 Q 1𝑜 <N 𝑛) (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈) → 𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))
4838, 39, 40, 33, 41, 46, 47syl231anc 1154 . . . . . . . . . 10 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → 𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))
49 df-rex 2306 . . . . . . . . . 10 (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈) ↔ 𝑚(𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)))
5048, 49sylib 127 . . . . . . . . 9 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → 𝑚(𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)))
5136, 50jca 290 . . . . . . . 8 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) 𝑚(𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))))
52 19.42v 1783 . . . . . . . 8 (𝑚((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))) ↔ ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) 𝑚(𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))))
5351, 52sylibr 137 . . . . . . 7 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → 𝑚((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))))
54 simprrl 491 . . . . . . . . . . . 12 (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))) → (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿)
55 eleq1 2097 . . . . . . . . . . . . . . . . 17 (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) → (𝑎 𝐿 ↔ (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿))
5655anbi1d 438 . . . . . . . . . . . . . . . 16 (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) → ((𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈) ↔ ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)))
5756anbi2d 437 . . . . . . . . . . . . . . 15 (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) → ((𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)) ↔ (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))))
5857anbi2d 437 . . . . . . . . . . . . . 14 (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) → (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))) ↔ ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)))))
5958ceqsexgv 2667 . . . . . . . . . . . . 13 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 → (𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)))) ↔ ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)))))
6059biimprcd 149 . . . . . . . . . . . 12 (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))) → ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))))))
6154, 60mpd 13 . . . . . . . . . . 11 (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))) → 𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)))))
62 simprrr 492 . . . . . . . . . . 11 (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))) → (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)
63 eleq1 2097 . . . . . . . . . . . . . . . . . 18 (𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) → (𝑏 𝑈 ↔ (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))
6463anbi2d 437 . . . . . . . . . . . . . . . . 17 (𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) → ((𝑎 𝐿 𝑏 𝑈) ↔ (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)))
6564anbi2d 437 . . . . . . . . . . . . . . . 16 (𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) → ((𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)) ↔ (𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))))
6665anbi2d 437 . . . . . . . . . . . . . . 15 (𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) → (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))) ↔ ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)))))
6766anbi2d 437 . . . . . . . . . . . . . 14 (𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) → ((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) ↔ (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))))))
6867exbidv 1703 . . . . . . . . . . . . 13 (𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) → (𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) ↔ 𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))))))
6968ceqsexgv 2667 . . . . . . . . . . . 12 ((x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈 → (𝑏(𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))))) ↔ 𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))))))
7069biimprcd 149 . . . . . . . . . . 11 (𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈)))) → ((x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈𝑏(𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))))))
7161, 62, 70sylc 56 . . . . . . . . . 10 (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))) → 𝑏(𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))))))
72 19.42v 1783 . . . . . . . . . . 11 (𝑎(𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))))) ↔ (𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))))))
7372exbii 1493 . . . . . . . . . 10 (𝑏𝑎(𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))))) ↔ 𝑏(𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑎(𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))))))
7471, 73sylibr 137 . . . . . . . . 9 (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))) → 𝑏𝑎(𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))))))
75 simprrl 491 . . . . . . . . . . . . . 14 (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))) → 𝑎 𝐿)
7675adantl 262 . . . . . . . . . . . . 13 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → 𝑎 𝐿)
77 simprrr 492 . . . . . . . . . . . . . . 15 (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))) → 𝑏 𝑈)
7877adantl 262 . . . . . . . . . . . . . 14 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → 𝑏 𝑈)
79 simpl 102 . . . . . . . . . . . . . . 15 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))))
80 simprl2 949 . . . . . . . . . . . . . . . 16 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → 𝑞 Q)
81 simprl3 950 . . . . . . . . . . . . . . . 16 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → (𝑞 +Q 𝑞) <Q 𝑃)
8280, 81jca 290 . . . . . . . . . . . . . . 15 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))
83 simprl1 948 . . . . . . . . . . . . . . . 16 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → x Q)
84 simprrl 491 . . . . . . . . . . . . . . . 16 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → 𝑚 𝜔)
8583, 84jca 290 . . . . . . . . . . . . . . 15 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → (x Q 𝑚 𝜔))
86 prarloclemcalc 6484 . . . . . . . . . . . . . . 15 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (x Q 𝑚 𝜔))) → 𝑏 <Q (𝑎 +Q 𝑃))
8779, 82, 85, 86syl12anc 1132 . . . . . . . . . . . . . 14 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → 𝑏 <Q (𝑎 +Q 𝑃))
8878, 87jca 290 . . . . . . . . . . . . 13 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
8976, 88jca 290 . . . . . . . . . . . 12 (((𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → (𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
9089ancom1s 503 . . . . . . . . . . 11 (((𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞))) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈)))) → (𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
9190anasss 379 . . . . . . . . . 10 ((𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))))) → (𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
92912eximi 1489 . . . . . . . . 9 (𝑏𝑎(𝑏 = (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) (𝑎 = (x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) ((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 (𝑎 𝐿 𝑏 𝑈))))) → 𝑏𝑎(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
9374, 92syl 14 . . . . . . . 8 (((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))) → 𝑏𝑎(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
9493exlimiv 1486 . . . . . . 7 (𝑚((x Q 𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃) (𝑚 𝜔 ((x +Q0 ([⟨𝑚, 1𝑜⟩] ~Q0 ·Q0 𝑞)) 𝐿 (x +Q ([⟨(𝑚 +𝑜 2𝑜), 1𝑜⟩] ~Q ·Q 𝑞)) 𝑈))) → 𝑏𝑎(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
9553, 94syl 14 . . . . . 6 ((((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → 𝑏𝑎(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
9695exlimivv 1773 . . . . 5 (𝑞𝑛(((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → 𝑏𝑎(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
9796exlimivv 1773 . . . 4 (xy𝑞𝑛(((x Q x 𝐿) (y Q y 𝑈) ((⟨𝐿, 𝑈 P 𝑃 Q) (𝑞 Q (𝑞 +Q 𝑞) <Q 𝑃))) (𝑛 N (1𝑜 <N 𝑛 y <Q (x +Q ([⟨𝑛, 1𝑜⟩] ~Q ·Q 𝑞))))) → 𝑏𝑎(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
9817, 30, 973syl 17 . . 3 ((⟨𝐿, 𝑈 P 𝑃 Q) → 𝑏𝑎(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
99 excom 1551 . . 3 (𝑏𝑎(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))) ↔ 𝑎𝑏(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
10098, 99sylib 127 . 2 ((⟨𝐿, 𝑈 P 𝑃 Q) → 𝑎𝑏(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
101 19.42v 1783 . . . . 5 (𝑏(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))) ↔ (𝑎 𝐿 𝑏(𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
102 df-rex 2306 . . . . . 6 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃) ↔ 𝑏(𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
103102anbi2i 430 . . . . 5 ((𝑎 𝐿 𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃)) ↔ (𝑎 𝐿 𝑏(𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))))
104101, 103bitr4i 176 . . . 4 (𝑏(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))) ↔ (𝑎 𝐿 𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
105104exbii 1493 . . 3 (𝑎𝑏(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))) ↔ 𝑎(𝑎 𝐿 𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
106 df-rex 2306 . . 3 (𝑎 𝐿 𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃) ↔ 𝑎(𝑎 𝐿 𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃)))
107105, 106bitr4i 176 . 2 (𝑎𝑏(𝑎 𝐿 (𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))) ↔ 𝑎 𝐿 𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))
108100, 107sylib 127 1 ((⟨𝐿, 𝑈 P 𝑃 Q) → 𝑎 𝐿 𝑏 𝑈 𝑏 <Q (𝑎 +Q 𝑃))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242  wex 1378   wcel 1390  wrex 2301  cop 3370   class class class wbr 3755  𝜔com 4256  (class class class)co 5455  1𝑜c1o 5933  2𝑜c2o 5934   +𝑜 coa 5937  [cec 6040  Ncnpi 6256   <N clti 6259   ~Q ceq 6263  Qcnq 6264   +Q cplq 6266   ·Q cmq 6267   <Q cltq 6269   ~Q0 ceq0 6270   +Q0 cplq0 6273   ·Q0 cmq0 6274  Pcnp 6275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448
This theorem is referenced by:  prarloc2  6486  addlocpr  6519  prmuloc  6545  ltaddpr  6569  ltexprlemloc  6579  ltexprlemrl  6582  ltexprlemru  6584
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