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Theorem recexprlem1ssu 6689
Description: The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 6693. (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 6611 . . . 4 (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
21abeq2i 2148 . . 3 (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
3 prop 6530 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
4 prmuloc2 6622 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q 𝑤) ∈ (2nd𝐴))
53, 4sylan 267 . . . . 5 ((𝐴P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q 𝑤) ∈ (2nd𝐴))
6 prnminu 6544 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
73, 6sylan 267 . . . . . . 7 ((𝐴P ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
87ad2ant2rl 480 . . . . . 6 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
9 simp3 906 . . . . . . . . . . 11 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑧 <Q (𝑣 ·Q 𝑤))
10 simp2l 930 . . . . . . . . . . . 12 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ (1st𝐴))
11 elprnql 6536 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (1st𝐴)) → 𝑣Q)
123, 11sylan 267 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑣 ∈ (1st𝐴)) → 𝑣Q)
1312ad2ant2r 478 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → 𝑣Q)
14133adant3 924 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣Q)
15 simp1r 929 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 1Q <Q 𝑤)
16 ltrelnq 6420 . . . . . . . . . . . . . . . . . 18 <Q ⊆ (Q × Q)
1716brel 4355 . . . . . . . . . . . . . . . . 17 (1Q <Q 𝑤 → (1QQ𝑤Q))
1817simprd 107 . . . . . . . . . . . . . . . 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 6447 . . . . . . . . . . . . . . . 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 6439 . . . . . . . . . . . . . . 15 ((𝑣Q𝑤Q ∧ (*Q𝑤) ∈ Q) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) = (𝑣 ·Q (𝑤 ·Q (*Q𝑤))))
2314, 19, 21, 22syl3anc 1135 . . . . . . . . . . . . . 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 6448 . . . . . . . . . . . . . . . 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 5491 . . . . . . . . . . . . . 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 6444 . . . . . . . . . . . . . . 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 2076 . . . . . . . . . . . . 13 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) = 𝑣)
3029eleq1d 2106 . . . . . . . . . . . 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 156 . . . . . . . . . . 11 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ (1st𝐴))
32 ltrnqi 6476 . . . . . . . . . . . . 13 (𝑧 <Q (𝑣 ·Q 𝑤) → (*Q‘(𝑣 ·Q 𝑤)) <Q (*Q𝑧))
33 ltmnqg 6456 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( ·Q 𝑓) <Q ( ·Q 𝑔)))
3433adantl 262 . . . . . . . . . . . . . 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 6431 . . . . . . . . . . . . . . . 16 ((𝑣Q𝑤Q) → (𝑣 ·Q 𝑤) ∈ Q)
3614, 19, 35syl2anc 391 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑣 ·Q 𝑤) ∈ Q)
37 recclnq 6447 . . . . . . . . . . . . . . 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 4355 . . . . . . . . . . . . . . . . 17 (𝑧 <Q (𝑣 ·Q 𝑤) → (𝑧Q ∧ (𝑣 ·Q 𝑤) ∈ Q))
4039simpld 105 . . . . . . . . . . . . . . . 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 6447 . . . . . . . . . . . . . . 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 6438 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
4544adantl 262 . . . . . . . . . . . . . 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 5633 . . . . . . . . . . . . 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 143 . . . . . . . . . . . 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 6421 . . . . . . . . . . . . . . . . 17 1QQ
49 mulidnq 6444 . . . . . . . . . . . . . . . . 17 (1QQ → (1Q ·Q 1Q) = 1Q)
5048, 49ax-mp 7 . . . . . . . . . . . . . . . 16 (1Q ·Q 1Q) = 1Q
51 mulcomnqg 6438 . . . . . . . . . . . . . . . . . . . . 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 398 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
53 recidnq 6448 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = 1Q)
5452, 53eqtr3d 2074 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ·Q 𝑤) ∈ Q → ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) = 1Q)
5554, 24oveqan12d 5494 . . . . . . . . . . . . . . . . . 18 (((𝑣 ·Q 𝑤) ∈ Q𝑤Q) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) ·Q (𝑤 ·Q (*Q𝑤))) = (1Q ·Q 1Q))
5636, 19, 55syl2anc 391 . . . . . . . . . . . . . . . . 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 6439 . . . . . . . . . . . . . . . . . . 19 ((𝑓Q𝑔QQ) → ((𝑓 ·Q 𝑔) ·Q ) = (𝑓 ·Q (𝑔 ·Q )))
5857adantl 262 . . . . . . . . . . . . . . . . . 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 6431 . . . . . . . . . . . . . . . . . . 19 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
6059adantl 262 . . . . . . . . . . . . . . . . . 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 5648 . . . . . . . . . . . . . . . . 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 2074 . . . . . . . . . . . . . . . 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 2087 . . . . . . . . . . . . . . 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 5617 . . . . . . . . . . . . . . . 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 5617 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ Q)
66 recmulnqg 6446 . . . . . . . . . . . . . . . 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 391 . . . . . . . . . . . . . . 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 156 . . . . . . . . . . . . . 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 2106 . . . . . . . . . . . . 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 147 . . . . . . . . . . . 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 3764 . . . . . . . . . . . . . . . 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 5141 . . . . . . . . . . . . . . . . 17 (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (*Q𝑦) = (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)))
7372eleq1d 2106 . . . . . . . . . . . . . . . 16 (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st𝐴)))
7471, 73anbi12d 442 . . . . . . . . . . . . . . 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 2638 . . . . . . . . . . . . . 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 6678 . . . . . . . . . . . . 13 (((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q𝑦) ∈ (1st𝐴)))
7976, 78syl6ibr 151 . . . . . . . . . . . 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 279 . . . . . . . . . . 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 409 . . . . . . . . . 10 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵))
82 mulidnq 6444 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = 𝑤)
83 mulcomnqg 6438 . . . . . . . . . . . . . . 15 ((𝑤Q ∧ 1QQ) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8448, 83mpan2 401 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8582, 84eqtr3d 2074 . . . . . . . . . . . . 13 (𝑤Q𝑤 = (1Q ·Q 𝑤))
8685adantl 262 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → 𝑤 = (1Q ·Q 𝑤))
87 recidnq 6448 . . . . . . . . . . . . . 14 (𝑧Q → (𝑧 ·Q (*Q𝑧)) = 1Q)
8887oveq1d 5490 . . . . . . . . . . . . 13 (𝑧Q → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
8988adantr 261 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
90 mulassnqg 6439 . . . . . . . . . . . . . 14 ((𝑧Q ∧ (*Q𝑧) ∈ Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9142, 90syl3an2 1169 . . . . . . . . . . . . 13 ((𝑧Q𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
92913anidm12 1192 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9386, 89, 923eqtr2d 2078 . . . . . . . . . . 11 ((𝑧Q𝑤Q) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9441, 19, 93syl2anc 391 . . . . . . . . . 10 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
95 oveq2 5483 . . . . . . . . . . . 12 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9695eqeq2d 2051 . . . . . . . . . . 11 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))))
9796rspcev 2653 . . . . . . . . . 10 ((((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵) ∧ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥))
9881, 94, 97syl2anc 391 . . . . . . . . 9 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥))
99983expia 1106 . . . . . . . 8 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → (𝑧 <Q (𝑣 ·Q 𝑤) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
10099reximdv 2417 . . . . . . 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 6687 . . . . . . . . 9 (𝐴P𝐵P)
102 df-imp 6524 . . . . . . . . . 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 6568 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
104101, 103mpdan 398 . . . . . . . 8 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
105104ad2antrr 457 . . . . . . 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 158 . . . . . 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 2434 . . . 4 ((𝐴P ∧ 1Q <Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)))
109108ex 108 . . 3 (𝐴P → (1Q <Q 𝑤𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
1102, 109syl5bi 141 . 2 (𝐴P → (𝑤 ∈ (2nd ‘1P) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
111110ssrdv 2948 1 (𝐴P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wex 1381  wcel 1393  {cab 2026  wrex 2304  wss 2914  cop 3375   class class class wbr 3761  cfv 4865  (class class class)co 5475  1st c1st 5728  2nd c2nd 5729  Qcnq 6335  1Qc1q 6336   ·Q cmq 6338  *Qcrq 6339   <Q cltq 6340  Pcnp 6346  1Pc1p 6347   ·P cmp 6349
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-recs 5883  df-irdg 5920  df-1o 5964  df-2o 5965  df-oadd 5968  df-omul 5969  df-er 6069  df-ec 6071  df-qs 6075  df-ni 6359  df-pli 6360  df-mi 6361  df-lti 6362  df-plpq 6399  df-mpq 6400  df-enq 6402  df-nqqs 6403  df-plqqs 6404  df-mqqs 6405  df-1nqqs 6406  df-rq 6407  df-ltnqqs 6408  df-enq0 6479  df-nq0 6480  df-0nq0 6481  df-plq0 6482  df-mq0 6483  df-inp 6521  df-i1p 6522  df-imp 6524
This theorem is referenced by:  recexprlemex  6692
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