ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexprlem1ssu GIF version

Theorem recexprlem1ssu 7094
Description: The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 7098. (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 7016 . . . 4 (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
21abeq2i 2193 . . 3 (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
3 prop 6935 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
4 prmuloc2 7027 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q 𝑤) ∈ (2nd𝐴))
53, 4sylan 277 . . . . 5 ((𝐴P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q 𝑤) ∈ (2nd𝐴))
6 prnminu 6949 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
73, 6sylan 277 . . . . . . 7 ((𝐴P ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
87ad2ant2rl 495 . . . . . 6 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
9 simp3 941 . . . . . . . . . . 11 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑧 <Q (𝑣 ·Q 𝑤))
10 simp2l 965 . . . . . . . . . . . 12 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ (1st𝐴))
11 elprnql 6941 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (1st𝐴)) → 𝑣Q)
123, 11sylan 277 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑣 ∈ (1st𝐴)) → 𝑣Q)
1312ad2ant2r 493 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → 𝑣Q)
14133adant3 959 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣Q)
15 simp1r 964 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 1Q <Q 𝑤)
16 ltrelnq 6825 . . . . . . . . . . . . . . . . . 18 <Q ⊆ (Q × Q)
1716brel 4446 . . . . . . . . . . . . . . . . 17 (1Q <Q 𝑤 → (1QQ𝑤Q))
1817simprd 112 . . . . . . . . . . . . . . . 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 6852 . . . . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . 15 ((𝑣Q𝑤Q ∧ (*Q𝑤) ∈ Q) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) = (𝑣 ·Q (𝑤 ·Q (*Q𝑤))))
2314, 19, 21, 22syl3anc 1170 . . . . . . . . . . . . . 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 6853 . . . . . . . . . . . . . . . 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 5605 . . . . . . . . . . . . . 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 6849 . . . . . . . . . . . . . . 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 2119 . . . . . . . . . . . . 13 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) = 𝑣)
3029eleq1d 2151 . . . . . . . . . . . 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 165 . . . . . . . . . . 11 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ (1st𝐴))
32 ltrnqi 6881 . . . . . . . . . . . . 13 (𝑧 <Q (𝑣 ·Q 𝑤) → (*Q‘(𝑣 ·Q 𝑤)) <Q (*Q𝑧))
33 ltmnqg 6861 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( ·Q 𝑓) <Q ( ·Q 𝑔)))
3433adantl 271 . . . . . . . . . . . . . 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 6836 . . . . . . . . . . . . . . . 16 ((𝑣Q𝑤Q) → (𝑣 ·Q 𝑤) ∈ Q)
3614, 19, 35syl2anc 403 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑣 ·Q 𝑤) ∈ Q)
37 recclnq 6852 . . . . . . . . . . . . . . 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 4446 . . . . . . . . . . . . . . . . 17 (𝑧 <Q (𝑣 ·Q 𝑤) → (𝑧Q ∧ (𝑣 ·Q 𝑤) ∈ Q))
4039simpld 110 . . . . . . . . . . . . . . . 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 6852 . . . . . . . . . . . . . . 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 6843 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
4544adantl 271 . . . . . . . . . . . . . 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 5747 . . . . . . . . . . . . 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 152 . . . . . . . . . . . 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 6826 . . . . . . . . . . . . . . . . 17 1QQ
49 mulidnq 6849 . . . . . . . . . . . . . . . . 17 (1QQ → (1Q ·Q 1Q) = 1Q)
5048, 49ax-mp 7 . . . . . . . . . . . . . . . 16 (1Q ·Q 1Q) = 1Q
51 mulcomnqg 6843 . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
53 recidnq 6853 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = 1Q)
5452, 53eqtr3d 2117 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ·Q 𝑤) ∈ Q → ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) = 1Q)
5554, 24oveqan12d 5608 . . . . . . . . . . . . . . . . . 18 (((𝑣 ·Q 𝑤) ∈ Q𝑤Q) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) ·Q (𝑤 ·Q (*Q𝑤))) = (1Q ·Q 1Q))
5636, 19, 55syl2anc 403 . . . . . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . . . . . 19 ((𝑓Q𝑔QQ) → ((𝑓 ·Q 𝑔) ·Q ) = (𝑓 ·Q (𝑔 ·Q )))
5857adantl 271 . . . . . . . . . . . . . . . . . 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 6836 . . . . . . . . . . . . . . . . . . 19 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
6059adantl 271 . . . . . . . . . . . . . . . . . 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 5762 . . . . . . . . . . . . . . . . 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 2117 . . . . . . . . . . . . . . . 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 2130 . . . . . . . . . . . . . . 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 5731 . . . . . . . . . . . . . . . 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 5731 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ Q)
66 recmulnqg 6851 . . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . . 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 165 . . . . . . . . . . . . . 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 2151 . . . . . . . . . . . . 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 156 . . . . . . . . . . . 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 3814 . . . . . . . . . . . . . . . 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 5251 . . . . . . . . . . . . . . . . 17 (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (*Q𝑦) = (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)))
7372eleq1d 2151 . . . . . . . . . . . . . . . 16 (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st𝐴)))
7471, 73anbi12d 457 . . . . . . . . . . . . . . 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 2697 . . . . . . . . . . . . . 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 7083 . . . . . . . . . . . . 13 (((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q𝑦) ∈ (1st𝐴)))
7976, 78syl6ibr 160 . . . . . . . . . . . 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 289 . . . . . . . . . . 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 424 . . . . . . . . . 10 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵))
82 mulidnq 6849 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = 𝑤)
83 mulcomnqg 6843 . . . . . . . . . . . . . . 15 ((𝑤Q ∧ 1QQ) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8448, 83mpan2 416 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8582, 84eqtr3d 2117 . . . . . . . . . . . . 13 (𝑤Q𝑤 = (1Q ·Q 𝑤))
8685adantl 271 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → 𝑤 = (1Q ·Q 𝑤))
87 recidnq 6853 . . . . . . . . . . . . . 14 (𝑧Q → (𝑧 ·Q (*Q𝑧)) = 1Q)
8887oveq1d 5604 . . . . . . . . . . . . 13 (𝑧Q → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
8988adantr 270 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
90 mulassnqg 6844 . . . . . . . . . . . . . 14 ((𝑧Q ∧ (*Q𝑧) ∈ Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9142, 90syl3an2 1204 . . . . . . . . . . . . 13 ((𝑧Q𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
92913anidm12 1227 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9386, 89, 923eqtr2d 2121 . . . . . . . . . . 11 ((𝑧Q𝑤Q) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9441, 19, 93syl2anc 403 . . . . . . . . . 10 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
95 oveq2 5597 . . . . . . . . . . . 12 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9695eqeq2d 2094 . . . . . . . . . . 11 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))))
9796rspcev 2712 . . . . . . . . . 10 ((((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵) ∧ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥))
9881, 94, 97syl2anc 403 . . . . . . . . 9 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥))
99983expia 1141 . . . . . . . 8 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → (𝑧 <Q (𝑣 ·Q 𝑤) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
10099reximdv 2468 . . . . . . 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 7092 . . . . . . . . 9 (𝐴P𝐵P)
102 df-imp 6929 . . . . . . . . . 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 6973 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
104101, 103mpdan 412 . . . . . . . 8 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
105104ad2antrr 472 . . . . . . 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 167 . . . . . 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 2487 . . . 4 ((𝐴P ∧ 1Q <Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)))
109108ex 113 . . 3 (𝐴P → (1Q <Q 𝑤𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
1102, 109syl5bi 150 . 2 (𝐴P → (𝑤 ∈ (2nd ‘1P) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
111110ssrdv 3016 1 (𝐴P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wrex 2354  wss 2984  cop 3425   class class class wbr 3811  cfv 4967  (class class class)co 5589  1st c1st 5842  2nd c2nd 5843  Qcnq 6740  1Qc1q 6741   ·Q cmq 6743  *Qcrq 6744   <Q cltq 6745  Pcnp 6751  1Pc1p 6752   ·P cmp 6754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4079  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-irdg 6065  df-1o 6111  df-2o 6112  df-oadd 6115  df-omul 6116  df-er 6220  df-ec 6222  df-qs 6226  df-ni 6764  df-pli 6765  df-mi 6766  df-lti 6767  df-plpq 6804  df-mpq 6805  df-enq 6807  df-nqqs 6808  df-plqqs 6809  df-mqqs 6810  df-1nqqs 6811  df-rq 6812  df-ltnqqs 6813  df-enq0 6884  df-nq0 6885  df-0nq0 6886  df-plq0 6887  df-mq0 6888  df-inp 6926  df-i1p 6927  df-imp 6929
This theorem is referenced by:  recexprlemex  7097
  Copyright terms: Public domain W3C validator