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Theorem recexprlem1ssu 7624
Description: The upper cut of one is a subset of the upper cut of 𝐴 ·P 𝐵. Lemma for recexpr 7628. (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 7546 . . . 4 (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
21abeq2i 2288 . . 3 (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
3 prop 7465 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
4 prmuloc2 7557 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q 𝑤) ∈ (2nd𝐴))
53, 4sylan 283 . . . . 5 ((𝐴P ∧ 1Q <Q 𝑤) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q 𝑤) ∈ (2nd𝐴))
6 prnminu 7479 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
73, 6sylan 283 . . . . . . 7 ((𝐴P ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
87ad2ant2rl 511 . . . . . 6 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → ∃𝑧 ∈ (2nd𝐴)𝑧 <Q (𝑣 ·Q 𝑤))
9 simp3 999 . . . . . . . . . . 11 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑧 <Q (𝑣 ·Q 𝑤))
10 simp2l 1023 . . . . . . . . . . . 12 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣 ∈ (1st𝐴))
11 elprnql 7471 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (1st𝐴)) → 𝑣Q)
123, 11sylan 283 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑣 ∈ (1st𝐴)) → 𝑣Q)
1312ad2ant2r 509 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → 𝑣Q)
14133adant3 1017 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑣Q)
15 simp1r 1022 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 1Q <Q 𝑤)
16 ltrelnq 7355 . . . . . . . . . . . . . . . . . 18 <Q ⊆ (Q × Q)
1716brel 4675 . . . . . . . . . . . . . . . . 17 (1Q <Q 𝑤 → (1QQ𝑤Q))
1817simprd 114 . . . . . . . . . . . . . . . 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 7382 . . . . . . . . . . . . . . . 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 7374 . . . . . . . . . . . . . . 15 ((𝑣Q𝑤Q ∧ (*Q𝑤) ∈ Q) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) = (𝑣 ·Q (𝑤 ·Q (*Q𝑤))))
2314, 19, 21, 22syl3anc 1238 . . . . . . . . . . . . . 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 7383 . . . . . . . . . . . . . . . 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 5885 . . . . . . . . . . . . . 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 7379 . . . . . . . . . . . . . . 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 2214 . . . . . . . . . . . . 13 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) = 𝑣)
3029eleq1d 2246 . . . . . . . . . . . 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 167 . . . . . . . . . . 11 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ (1st𝐴))
32 ltrnqi 7411 . . . . . . . . . . . . 13 (𝑧 <Q (𝑣 ·Q 𝑤) → (*Q‘(𝑣 ·Q 𝑤)) <Q (*Q𝑧))
33 ltmnqg 7391 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( ·Q 𝑓) <Q ( ·Q 𝑔)))
3433adantl 277 . . . . . . . . . . . . . 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 7366 . . . . . . . . . . . . . . . 16 ((𝑣Q𝑤Q) → (𝑣 ·Q 𝑤) ∈ Q)
3614, 19, 35syl2anc 411 . . . . . . . . . . . . . . 15 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → (𝑣 ·Q 𝑤) ∈ Q)
37 recclnq 7382 . . . . . . . . . . . . . . 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 4675 . . . . . . . . . . . . . . . . 17 (𝑧 <Q (𝑣 ·Q 𝑤) → (𝑧Q ∧ (𝑣 ·Q 𝑤) ∈ Q))
4039simpld 112 . . . . . . . . . . . . . . . 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 7382 . . . . . . . . . . . . . . 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 7373 . . . . . . . . . . . . . . 15 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
4544adantl 277 . . . . . . . . . . . . . 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 6038 . . . . . . . . . . . . 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, 46imbitrid 154 . . . . . . . . . . . 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 mulcomnqg 7373 . . . . . . . . . . . . . . . . . . . . 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)))
4937, 48mpdan 421 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)))
50 recidnq 7383 . . . . . . . . . . . . . . . . . . . 20 ((𝑣 ·Q 𝑤) ∈ Q → ((𝑣 ·Q 𝑤) ·Q (*Q‘(𝑣 ·Q 𝑤))) = 1Q)
5149, 50eqtr3d 2212 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ·Q 𝑤) ∈ Q → ((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) = 1Q)
5251, 24oveqan12d 5888 . . . . . . . . . . . . . . . . . 18 (((𝑣 ·Q 𝑤) ∈ Q𝑤Q) → (((*Q‘(𝑣 ·Q 𝑤)) ·Q (𝑣 ·Q 𝑤)) ·Q (𝑤 ·Q (*Q𝑤))) = (1Q ·Q 1Q))
5336, 19, 52syl2anc 411 . . . . . . . . . . . . . . . . 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))
54 mulassnqg 7374 . . . . . . . . . . . . . . . . . . 19 ((𝑓Q𝑔QQ) → ((𝑓 ·Q 𝑔) ·Q ) = (𝑓 ·Q (𝑔 ·Q )))
5554adantl 277 . . . . . . . . . . . . . . . . . 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 )))
56 mulclnq 7366 . . . . . . . . . . . . . . . . . . 19 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
5756adantl 277 . . . . . . . . . . . . . . . . . 18 ((((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) ∧ (𝑓Q𝑔Q)) → (𝑓 ·Q 𝑔) ∈ Q)
5838, 36, 19, 45, 55, 21, 57caov4d 6053 . . . . . . . . . . . . . . . . 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))))
5953, 58eqtr3d 2212 . . . . . . . . . . . . . . . 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))))
60 1nq 7356 . . . . . . . . . . . . . . . . 17 1QQ
61 mulidnq 7379 . . . . . . . . . . . . . . . . 17 (1QQ → (1Q ·Q 1Q) = 1Q)
6260, 61ax-mp 5 . . . . . . . . . . . . . . . 16 (1Q ·Q 1Q) = 1Q
6359, 62eqtr3di 2225 . . . . . . . . . . . . . . 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)
6457, 38, 19caovcld 6022 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) ∈ Q)
6557, 36, 21caovcld 6022 . . . . . . . . . . . . . . . 16 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((𝑣 ·Q 𝑤) ·Q (*Q𝑤)) ∈ Q)
66 recmulnqg 7381 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 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 167 . . . . . . . . . . . . . 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 2246 . . . . . . . . . . . . 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 158 . . . . . . . . . . . 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 4003 . . . . . . . . . . . . . . . 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 5511 . . . . . . . . . . . . . . . . 17 (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → (*Q𝑦) = (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)))
7372eleq1d 2246 . . . . . . . . . . . . . . . 16 (𝑦 = ((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘((*Q‘(𝑣 ·Q 𝑤)) ·Q 𝑤)) ∈ (1st𝐴)))
7471, 73anbi12d 473 . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . 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 7613 . . . . . . . . . . . . 13 (((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q ((*Q𝑧) ·Q 𝑤) ∧ (*Q𝑦) ∈ (1st𝐴)))
7976, 78syl6ibr 162 . . . . . . . . . . . 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 295 . . . . . . . . . . 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 433 . . . . . . . . . 10 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵))
82 mulidnq 7379 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = 𝑤)
83 mulcomnqg 7373 . . . . . . . . . . . . . . 15 ((𝑤Q ∧ 1QQ) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8460, 83mpan2 425 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8582, 84eqtr3d 2212 . . . . . . . . . . . . 13 (𝑤Q𝑤 = (1Q ·Q 𝑤))
8685adantl 277 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → 𝑤 = (1Q ·Q 𝑤))
87 recidnq 7383 . . . . . . . . . . . . . 14 (𝑧Q → (𝑧 ·Q (*Q𝑧)) = 1Q)
8887oveq1d 5884 . . . . . . . . . . . . 13 (𝑧Q → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
8988adantr 276 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
90 mulassnqg 7374 . . . . . . . . . . . . . 14 ((𝑧Q ∧ (*Q𝑧) ∈ Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9142, 90syl3an2 1272 . . . . . . . . . . . . 13 ((𝑧Q𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
92913anidm12 1295 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9386, 89, 923eqtr2d 2216 . . . . . . . . . . 11 ((𝑧Q𝑤Q) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9441, 19, 93syl2anc 411 . . . . . . . . . 10 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
95 oveq2 5877 . . . . . . . . . . . 12 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9695eqeq2d 2189 . . . . . . . . . . 11 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))))
9796rspcev 2841 . . . . . . . . . 10 ((((*Q𝑧) ·Q 𝑤) ∈ (2nd𝐵) ∧ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥))
9881, 94, 97syl2anc 411 . . . . . . . . 9 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴)) ∧ 𝑧 <Q (𝑣 ·Q 𝑤)) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥))
99983expia 1205 . . . . . . . 8 (((𝐴P ∧ 1Q <Q 𝑤) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q 𝑤) ∈ (2nd𝐴))) → (𝑧 <Q (𝑣 ·Q 𝑤) → ∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
10099reximdv 2578 . . . . . . 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 7622 . . . . . . . . 9 (𝐴P𝐵P)
102 df-imp 7459 . . . . . . . . . 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, 56genpelvu 7503 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
104101, 103mpdan 421 . . . . . . . 8 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑥 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑥)))
105104ad2antrr 488 . . . . . . 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 169 . . . . . 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 2599 . . . 4 ((𝐴P ∧ 1Q <Q 𝑤) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)))
109108ex 115 . . 3 (𝐴P → (1Q <Q 𝑤𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
1102, 109biimtrid 152 . 2 (𝐴P → (𝑤 ∈ (2nd ‘1P) → 𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵))))
111110ssrdv 3161 1 (𝐴P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wrex 2456  wss 3129  cop 3594   class class class wbr 4000  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Qcnq 7270  1Qc1q 7271   ·Q cmq 7273  *Qcrq 7274   <Q cltq 7275  Pcnp 7281  1Pc1p 7282   ·P cmp 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-imp 7459
This theorem is referenced by:  recexprlemex  7627
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