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Theorem recexprlem1ssu 6790
Description: The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 6794. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
Assertion
Ref Expression
recexprlem1ssu (𝐴P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem recexprlem1ssu
Dummy variables 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1pru 6712 . . . 4 (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
21abeq2i 2164 . . 3 (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
3 prop 6631 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
4 prmuloc2 6723 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q 𝑤) ∈ (2nd𝐴))
53, 4sylan 271 . . . . 5 ((𝐴P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q 𝑤) ∈ (2nd𝐴))
6 prnminu 6645 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
73, 6sylan 271 . . . . . . 7 ((𝐴P ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
87ad2ant2rl 488 . . . . . 6 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
9 simp3 917 . . . . . . . . . . 11 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑧 <Q (𝑣 ·Q 𝑤))
10 simp2l 941 . . . . . . . . . . . 12 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ (1st𝐴))
11 elprnql 6637 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (1st𝐴)) → 𝑣Q)
123, 11sylan 271 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑣 ∈ (1st𝐴)) → 𝑣Q)
1312ad2ant2r 486 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → 𝑣Q)
14133adant3 935 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣Q)
15 simp1r 940 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 1Q <Q 𝑤)
16 ltrelnq 6521 . . . . . . . . . . . . . . . . . 18 <Q ⊆ (Q × Q)
1716brel 4420 . . . . . . . . . . . . . . . . 17 (1Q <Q 𝑤 → (1QQ𝑤Q))
1817simprd 111 . . . . . . . . . . . . . . . 16 (1Q <Q 𝑤𝑤Q)
1915, 18syl 14 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑤Q)
20 recclnq 6548 . . . . . . . . . . . . . . . 16 (𝑤Q → (*Q𝑤) ∈ Q)
2119, 20syl 14 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (*Q𝑤) ∈ Q)
22 mulassnqg 6540 . . . . . . . . . . . . . . 15 ((𝑣Q𝑤Q ∧ (*Q𝑤) ∈ Q) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) = (𝑣 ·Q (𝑤 ·Q (*Q𝑤))))
2314, 19, 21, 22syl3anc 1146 . . . . . . . . . . . . . 14 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) = (𝑣 ·Q (𝑤 ·Q (*Q𝑤))))
24 recidnq 6549 . . . . . . . . . . . . . . . 16 (𝑤Q → (𝑤 ·Q (*Q𝑤)) = 1Q)
2519, 24syl 14 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑤 ·Q (*Q𝑤)) = 1Q)
2625oveq2d 5556 . . . . . . . . . . . . . 14 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑣 ·Q (𝑤 ·Q (*Q𝑤))) = (𝑣 ·Q 1Q))
27 mulidnq 6545 . . . . . . . . . . . . . . 15 (𝑣Q → (𝑣 ·Q 1Q) = 𝑣)
2814, 27syl 14 . . . . . . . . . . . . . 14 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑣 ·Q 1Q) = 𝑣)
2923, 26, 283eqtrd 2092 . . . . . . . . . . . . 13 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) = 𝑣)
3029eleq1d 2122 . . . . . . . . . . . 12 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ (1st𝐴) ↔ 𝑣 ∈ (1st𝐴)))
3110, 30mpbird 160 . . . . . . . . . . 11 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ (1st𝐴))
32 ltrnqi 6577 . . . . . . . . . . . . 13 (𝑧 <Q (𝑣 ·Q 𝑤) → (*Q‘(𝑣 ·Q 𝑤)) <Q (*Q𝑧))
33 ltmnqg 6557 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( ·Q 𝑓) <Q ( ·Q 𝑔)))
3433adantl 266 . . . . . . . . . . . . . 14 ((((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( ·Q 𝑓) <Q ( ·Q 𝑔)))
35 mulclnq 6532 . . . . . . . . . . . . . . . 16 ((𝑣Q𝑤Q) → (𝑣 ·Q 𝑤) ∈ Q)
3614, 19, 35syl2anc 397 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑣 ·Q 𝑤) ∈ Q)
37 recclnq 6548 . . . . . . . . . . . . . . 15 ((𝑣 ·Q 𝑤) ∈ Q → (*Q‘(𝑣 ·Q 𝑤)) ∈ Q)
3836, 37syl 14 . . . . . . . . . . . . . 14 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (*Q‘(𝑣 ·Q 𝑤)) ∈ Q)
3916brel 4420 . . . . . . . . . . . . . . . . 17 (𝑧 <Q (𝑣 ·Q 𝑤) → (𝑧Q ∧ (𝑣 ·Q 𝑤) ∈ Q))
4039simpld 109 . . . . . . . . . . . . . . . 16 (𝑧 <Q (𝑣 ·Q 𝑤) → 𝑧Q)
419, 40syl 14 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑧Q)
42 recclnq 6548 . . . . . . . . . . . . . . 15 (𝑧Q → (*Q𝑧) ∈ Q)
4341, 42syl 14 . . . . . . . . . . . . . 14 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (*Q𝑧) ∈ Q)
44 mulcomnqg 6539 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
4544adantl 266 . . . . . . . . . . . . . 14 ((((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) ∧ (𝑓Q𝑔Q)) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
4634, 38, 43, 19, 45caovord2d 5698 . . . . . . . . . . . . 13 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q‘(𝑣 ·Q 𝑤)) <Q (*Q𝑧) ↔ ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q𝑧) ·Q 𝑤)))
4732, 46syl5ib 147 . . . . . . . . . . . 12 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑧 <Q (𝑣 ·Q 𝑤) → ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q𝑧) ·Q 𝑤)))
48 1nq 6522 . . . . . . . . . . . . . . . . 17 1QQ
49 mulidnq 6545 . . . . . . . . . . . . . . . . 17 (1QQ → (1Q ·Q 1Q) = 1Q)
5048, 49ax-mp 7 . . . . . . . . . . . . . . . 16 (1Q ·Q 1Q) = 1Q
51 mulcomnqg 6539 . . . . . . . . . . . . . . . . . . . . 21 (((𝑣 ·Q 𝑤) ∈ Q ∧ (*Q‘(𝑣 ·Q 𝑤)) ∈ Q) → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
5237, 51mpdan 406 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
53 recidnq 6549 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = 1Q)
5452, 53eqtr3d 2090 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ·Q 𝑤) ∈ Q → ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) = 1Q)
5554, 24oveqan12d 5559 . . . . . . . . . . . . . . . . . 18 (((𝑣 ·Q 𝑤) ∈ Q𝑤Q) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) ·Q (𝑤 ·Q (*Q𝑤))) = (1Q ·Q 1Q))
5636, 19, 55syl2anc 397 . . . . . . . . . . . . . . . . 17 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) ·Q (𝑤 ·Q (*Q𝑤))) = (1Q ·Q 1Q))
57 mulassnqg 6540 . . . . . . . . . . . . . . . . . . 19 ((𝑓Q𝑔QQ) → ((𝑓 ·Q 𝑔) ·Q ) = (𝑓 ·Q (𝑔 ·Q )))
5857adantl 266 . . . . . . . . . . . . . . . . . 18 ((((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) ∧ (𝑓Q𝑔QQ)) → ((𝑓 ·Q 𝑔) ·Q ) = (𝑓 ·Q (𝑔 ·Q )))
59 mulclnq 6532 . . . . . . . . . . . . . . . . . . 19 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
6059adantl 266 . . . . . . . . . . . . . . . . . 18 ((((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) ∧ (𝑓Q𝑔Q)) → (𝑓 ·Q 𝑔) ∈ Q)
6138, 36, 19, 45, 58, 21, 60caov4d 5713 . . . . . . . . . . . . . . . . 17 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) ·Q (𝑤 ·Q (*Q𝑤))) = (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ·Q ((𝑣 ·Q 𝑤) ·Q (*Q𝑤))))
6256, 61eqtr3d 2090 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (1Q ·Q 1Q) = (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ·Q ((𝑣 ·Q 𝑤) ·Q (*Q𝑤))))
6350, 62syl5reqr 2103 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ·Q ((𝑣 ·Q 𝑤) ·Q (*Q𝑤))) = 1Q)
6460, 38, 19caovcld 5682 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ∈ Q)
6560, 36, 21caovcld 5682 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ Q)
66 recmulnqg 6547 . . . . . . . . . . . . . . . 16 ((((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ∈ Q ∧ ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ Q) → ((*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) = ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ↔ (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ·Q ((𝑣 ·Q 𝑤) ·Q (*Q𝑤))) = 1Q))
6764, 65, 66syl2anc 397 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) = ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ↔ (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ·Q ((𝑣 ·Q 𝑤) ·Q (*Q𝑤))) = 1Q))
6863, 67mpbird 160 . . . . . . . . . . . . . 14 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) = ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)))
6968eleq1d 2122 . . . . . . . . . . . . 13 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st𝐴) ↔ ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ (1st𝐴)))
7069biimprd 151 . . . . . . . . . . . 12 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ (1st𝐴) → (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st𝐴)))
71 breq1 3795 . . . . . . . . . . . . . . . 16 (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (𝑦 <Q ((*Q𝑧) ·Q 𝑤) ↔ ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q𝑧) ·Q 𝑤)))
72 fveq2 5206 . . . . . . . . . . . . . . . . 17 (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (*Q𝑦) = (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)))
7372eleq1d 2122 . . . . . . . . . . . . . . . 16 (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st𝐴)))
7471, 73anbi12d 450 . . . . . . . . . . . . . . 15 (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → ((𝑦 <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st𝐴))))
7574spcegv 2658 . . . . . . . . . . . . . 14 (((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ∈ Q → ((((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st𝐴)) → ∃𝑦(𝑦 <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q𝑦) ∈ (1st𝐴))))
7664, 75syl 14 . . . . . . . . . . . . 13 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st𝐴)) → ∃𝑦(𝑦 <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q𝑦) ∈ (1st𝐴))))
77 recexpr.1 . . . . . . . . . . . . . 14 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
7877recexprlemelu 6779 . . . . . . . . . . . . 13 (((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q𝑦) ∈ (1st𝐴)))
7976, 78syl6ibr 155 . . . . . . . . . . . 12 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st𝐴)) → ((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵)))
8047, 70, 79syl2and 283 . . . . . . . . . . 11 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑧 <Q (𝑣 ·Q 𝑤) ∧ ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ (1st𝐴)) → ((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵)))
819, 31, 80mp2and 417 . . . . . . . . . 10 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵))
82 mulidnq 6545 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = 𝑤)
83 mulcomnqg 6539 . . . . . . . . . . . . . . 15 ((𝑤Q ∧ 1QQ) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8448, 83mpan2 409 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8582, 84eqtr3d 2090 . . . . . . . . . . . . 13 (𝑤Q𝑤 = (1Q ·Q 𝑤))
8685adantl 266 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → 𝑤 = (1Q ·Q 𝑤))
87 recidnq 6549 . . . . . . . . . . . . . 14 (𝑧Q → (𝑧 ·Q (*Q𝑧)) = 1Q)
8887oveq1d 5555 . . . . . . . . . . . . 13 (𝑧Q → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
8988adantr 265 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
90 mulassnqg 6540 . . . . . . . . . . . . . 14 ((𝑧Q ∧ (*Q𝑧) ∈ Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9142, 90syl3an2 1180 . . . . . . . . . . . . 13 ((𝑧Q𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
92913anidm12 1203 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9386, 89, 923eqtr2d 2094 . . . . . . . . . . 11 ((𝑧Q𝑤Q) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9441, 19, 93syl2anc 397 . . . . . . . . . 10 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
95 oveq2 5548 . . . . . . . . . . . 12 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9695eqeq2d 2067 . . . . . . . . . . 11 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))))
9796rspcev 2673 . . . . . . . . . 10 ((((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵) ∧ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥))
9881, 94, 97syl2anc 397 . . . . . . . . 9 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥))
99983expia 1117 . . . . . . . 8 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → (𝑧 <Q (𝑣 ·Q 𝑤) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
10099reximdv 2437 . . . . . . 7 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → (∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤) → ∃𝑧 ∈ (2nd𝐴)∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
10177recexprlempr 6788 . . . . . . . . 9 (𝐴P𝐵P)
102 df-imp 6625 . . . . . . . . . 10 ·P = (𝑦P, 𝑤P ↦ ⟨{𝑢Q ∣ ∃𝑓Q𝑔Q (𝑓 ∈ (1st𝑦) ∧ 𝑔 ∈ (1st𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}, {𝑢Q ∣ ∃𝑓Q𝑔Q (𝑓 ∈ (2nd𝑦) ∧ 𝑔 ∈ (2nd𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}⟩)
103102, 59genpelvu 6669 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
104101, 103mpdan 406 . . . . . . . 8 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
105104ad2antrr 465 . . . . . . 7 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
106100, 105sylibrd 162 . . . . . 6 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → (∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
1078, 106mpd 13 . . . . 5 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)))
1085, 107rexlimddv 2454 . . . 4 ((𝐴P ∧ 1Q <Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)))
109108ex 112 . . 3 (𝐴P → (1Q <Q 𝑤𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
1102, 109syl5bi 145 . 2 (𝐴P → (𝑤 ∈ (2nd ‘1P) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
111110ssrdv 2979 1 (𝐴P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wrex 2324  wss 2945  cop 3406   class class class wbr 3792  cfv 4930  (class class class)co 5540  1st c1st 5793  2nd c2nd 5794  Qcnq 6436  1Qc1q 6437   ·Q cmq 6439  *Qcrq 6440   <Q cltq 6441  Pcnp 6447  1Pc1p 6448   ·P cmp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-imp 6625
This theorem is referenced by:  recexprlemex  6793
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