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Theorem recexprlem1ssl 7093
Description: The lower cut of one is a subset of the lower 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
recexprlem1ssl (𝐴P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem recexprlem1ssl
Dummy variables 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1prl 7015 . . . 4 (1st ‘1P) = {𝑤𝑤 <Q 1Q}
21abeq2i 2193 . . 3 (𝑤 ∈ (1st ‘1P) ↔ 𝑤 <Q 1Q)
3 rec1nq 6855 . . . . . . 7 (*Q‘1Q) = 1Q
4 ltrnqi 6881 . . . . . . 7 (𝑤 <Q 1Q → (*Q‘1Q) <Q (*Q𝑤))
53, 4syl5eqbrr 3845 . . . . . 6 (𝑤 <Q 1Q → 1Q <Q (*Q𝑤))
6 prop 6935 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 prmuloc2 7027 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ 1Q <Q (*Q𝑤)) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))
86, 7sylan 277 . . . . . 6 ((𝐴P ∧ 1Q <Q (*Q𝑤)) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))
95, 8sylan2 280 . . . . 5 ((𝐴P𝑤 <Q 1Q) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))
10 prnmaxl 6948 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (1st𝐴)) → ∃𝑧 ∈ (1st𝐴)𝑣 <Q 𝑧)
116, 10sylan 277 . . . . . . 7 ((𝐴P𝑣 ∈ (1st𝐴)) → ∃𝑧 ∈ (1st𝐴)𝑣 <Q 𝑧)
1211ad2ant2r 493 . . . . . 6 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ∃𝑧 ∈ (1st𝐴)𝑣 <Q 𝑧)
13 elprnql 6941 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (1st𝐴)) → 𝑣Q)
146, 13sylan 277 . . . . . . . . . . . . 13 ((𝐴P𝑣 ∈ (1st𝐴)) → 𝑣Q)
1514ad2ant2r 493 . . . . . . . . . . . 12 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → 𝑣Q)
16153adant3 959 . . . . . . . . . . 11 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑣Q)
17 simp1r 964 . . . . . . . . . . . 12 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤 <Q 1Q)
18 ltrelnq 6825 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
1918brel 4446 . . . . . . . . . . . . 13 (𝑤 <Q 1Q → (𝑤Q ∧ 1QQ))
2019simpld 110 . . . . . . . . . . . 12 (𝑤 <Q 1Q𝑤Q)
2117, 20syl 14 . . . . . . . . . . 11 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤Q)
22 simp3 941 . . . . . . . . . . 11 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑣 <Q 𝑧)
23 simp2r 966 . . . . . . . . . . 11 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))
24 simpr 108 . . . . . . . . . . . 12 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)))
25 ltrnqi 6881 . . . . . . . . . . . . . 14 (𝑣 <Q 𝑧 → (*Q𝑧) <Q (*Q𝑣))
26 ltmnqg 6861 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( ·Q 𝑓) <Q ( ·Q 𝑔)))
2726adantl 271 . . . . . . . . . . . . . . 15 ((((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( ·Q 𝑓) <Q ( ·Q 𝑔)))
28 simprl 498 . . . . . . . . . . . . . . . 16 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → 𝑣 <Q 𝑧)
2918brel 4446 . . . . . . . . . . . . . . . . 17 (𝑣 <Q 𝑧 → (𝑣Q𝑧Q))
3029simprd 112 . . . . . . . . . . . . . . . 16 (𝑣 <Q 𝑧𝑧Q)
31 recclnq 6852 . . . . . . . . . . . . . . . 16 (𝑧Q → (*Q𝑧) ∈ Q)
3228, 30, 313syl 17 . . . . . . . . . . . . . . 15 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (*Q𝑧) ∈ Q)
33 recclnq 6852 . . . . . . . . . . . . . . . 16 (𝑣Q → (*Q𝑣) ∈ Q)
3433ad2antrr 472 . . . . . . . . . . . . . . 15 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (*Q𝑣) ∈ Q)
35 simplr 497 . . . . . . . . . . . . . . 15 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → 𝑤Q)
36 mulcomnqg 6843 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
3736adantl 271 . . . . . . . . . . . . . . 15 ((((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) ∧ (𝑓Q𝑔Q)) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
3827, 32, 34, 35, 37caovord2d 5747 . . . . . . . . . . . . . 14 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ((*Q𝑧) <Q (*Q𝑣) ↔ ((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤)))
3925, 38syl5ib 152 . . . . . . . . . . . . 13 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (𝑣 <Q 𝑧 → ((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤)))
40 1nq 6826 . . . . . . . . . . . . . . . . . 18 1QQ
41 mulidnq 6849 . . . . . . . . . . . . . . . . . 18 (1QQ → (1Q ·Q 1Q) = 1Q)
4240, 41ax-mp 7 . . . . . . . . . . . . . . . . 17 (1Q ·Q 1Q) = 1Q
43 mulcomnqg 6843 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣Q ∧ (*Q𝑣) ∈ Q) → (𝑣 ·Q (*Q𝑣)) = ((*Q𝑣) ·Q 𝑣))
4433, 43mpdan 412 . . . . . . . . . . . . . . . . . . . . 21 (𝑣Q → (𝑣 ·Q (*Q𝑣)) = ((*Q𝑣) ·Q 𝑣))
45 recidnq 6853 . . . . . . . . . . . . . . . . . . . . 21 (𝑣Q → (𝑣 ·Q (*Q𝑣)) = 1Q)
4644, 45eqtr3d 2117 . . . . . . . . . . . . . . . . . . . 20 (𝑣Q → ((*Q𝑣) ·Q 𝑣) = 1Q)
47 recidnq 6853 . . . . . . . . . . . . . . . . . . . 20 (𝑤Q → (𝑤 ·Q (*Q𝑤)) = 1Q)
4846, 47oveqan12d 5608 . . . . . . . . . . . . . . . . . . 19 ((𝑣Q𝑤Q) → (((*Q𝑣) ·Q 𝑣) ·Q (𝑤 ·Q (*Q𝑤))) = (1Q ·Q 1Q))
4948adantr 270 . . . . . . . . . . . . . . . . . 18 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (((*Q𝑣) ·Q 𝑣) ·Q (𝑤 ·Q (*Q𝑤))) = (1Q ·Q 1Q))
50 simpll 496 . . . . . . . . . . . . . . . . . . 19 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → 𝑣Q)
51 mulassnqg 6844 . . . . . . . . . . . . . . . . . . . 20 ((𝑓Q𝑔QQ) → ((𝑓 ·Q 𝑔) ·Q ) = (𝑓 ·Q (𝑔 ·Q )))
5251adantl 271 . . . . . . . . . . . . . . . . . . 19 ((((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) ∧ (𝑓Q𝑔QQ)) → ((𝑓 ·Q 𝑔) ·Q ) = (𝑓 ·Q (𝑔 ·Q )))
53 recclnq 6852 . . . . . . . . . . . . . . . . . . . 20 (𝑤Q → (*Q𝑤) ∈ Q)
5435, 53syl 14 . . . . . . . . . . . . . . . . . . 19 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (*Q𝑤) ∈ Q)
55 mulclnq 6836 . . . . . . . . . . . . . . . . . . . 20 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
5655adantl 271 . . . . . . . . . . . . . . . . . . 19 ((((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) ∧ (𝑓Q𝑔Q)) → (𝑓 ·Q 𝑔) ∈ Q)
5734, 50, 35, 37, 52, 54, 56caov4d 5762 . . . . . . . . . . . . . . . . . 18 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (((*Q𝑣) ·Q 𝑣) ·Q (𝑤 ·Q (*Q𝑤))) = (((*Q𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q𝑤))))
5849, 57eqtr3d 2117 . . . . . . . . . . . . . . . . 17 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (1Q ·Q 1Q) = (((*Q𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q𝑤))))
5942, 58syl5reqr 2130 . . . . . . . . . . . . . . . 16 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (((*Q𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q𝑤))) = 1Q)
60 mulclnq 6836 . . . . . . . . . . . . . . . . . . 19 (((*Q𝑣) ∈ Q𝑤Q) → ((*Q𝑣) ·Q 𝑤) ∈ Q)
6133, 60sylan 277 . . . . . . . . . . . . . . . . . 18 ((𝑣Q𝑤Q) → ((*Q𝑣) ·Q 𝑤) ∈ Q)
62 mulclnq 6836 . . . . . . . . . . . . . . . . . . 19 ((𝑣Q ∧ (*Q𝑤) ∈ Q) → (𝑣 ·Q (*Q𝑤)) ∈ Q)
6353, 62sylan2 280 . . . . . . . . . . . . . . . . . 18 ((𝑣Q𝑤Q) → (𝑣 ·Q (*Q𝑤)) ∈ Q)
64 recmulnqg 6851 . . . . . . . . . . . . . . . . . 18 ((((*Q𝑣) ·Q 𝑤) ∈ Q ∧ (𝑣 ·Q (*Q𝑤)) ∈ Q) → ((*Q‘((*Q𝑣) ·Q 𝑤)) = (𝑣 ·Q (*Q𝑤)) ↔ (((*Q𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q𝑤))) = 1Q))
6561, 63, 64syl2anc 403 . . . . . . . . . . . . . . . . 17 ((𝑣Q𝑤Q) → ((*Q‘((*Q𝑣) ·Q 𝑤)) = (𝑣 ·Q (*Q𝑤)) ↔ (((*Q𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q𝑤))) = 1Q))
6665adantr 270 . . . . . . . . . . . . . . . 16 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ((*Q‘((*Q𝑣) ·Q 𝑤)) = (𝑣 ·Q (*Q𝑤)) ↔ (((*Q𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q𝑤))) = 1Q))
6759, 66mpbird 165 . . . . . . . . . . . . . . 15 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (*Q‘((*Q𝑣) ·Q 𝑤)) = (𝑣 ·Q (*Q𝑤)))
6867eleq1d 2151 . . . . . . . . . . . . . 14 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ((*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴) ↔ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)))
6968biimprd 156 . . . . . . . . . . . . 13 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ((𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴) → (*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴)))
70 breq2 3815 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((*Q𝑣) ·Q 𝑤) → (((*Q𝑧) ·Q 𝑤) <Q 𝑦 ↔ ((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤)))
71 fveq2 5251 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((*Q𝑣) ·Q 𝑤) → (*Q𝑦) = (*Q‘((*Q𝑣) ·Q 𝑤)))
7271eleq1d 2151 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((*Q𝑣) ·Q 𝑤) → ((*Q𝑦) ∈ (2nd𝐴) ↔ (*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴)))
7370, 72anbi12d 457 . . . . . . . . . . . . . . . . 17 (𝑦 = ((*Q𝑣) ·Q 𝑤) → ((((*Q𝑧) ·Q 𝑤) <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ (((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤) ∧ (*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴))))
7473spcegv 2697 . . . . . . . . . . . . . . . 16 (((*Q𝑣) ·Q 𝑤) ∈ Q → ((((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤) ∧ (*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴)) → ∃𝑦(((*Q𝑧) ·Q 𝑤) <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
7561, 74syl 14 . . . . . . . . . . . . . . 15 ((𝑣Q𝑤Q) → ((((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤) ∧ (*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴)) → ∃𝑦(((*Q𝑧) ·Q 𝑤) <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
76 recexpr.1 . . . . . . . . . . . . . . . 16 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
7776recexprlemell 7082 . . . . . . . . . . . . . . 15 (((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵) ↔ ∃𝑦(((*Q𝑧) ·Q 𝑤) <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
7875, 77syl6ibr 160 . . . . . . . . . . . . . 14 ((𝑣Q𝑤Q) → ((((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤) ∧ (*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴)) → ((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵)))
7978adantr 270 . . . . . . . . . . . . 13 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ((((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤) ∧ (*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴)) → ((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵)))
8039, 69, 79syl2and 289 . . . . . . . . . . . 12 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ((𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) → ((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵)))
8124, 80mpd 13 . . . . . . . . . . 11 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵))
8216, 21, 22, 23, 81syl22anc 1171 . . . . . . . . . 10 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → ((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵))
83303ad2ant3 962 . . . . . . . . . . 11 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑧Q)
84 mulidnq 6849 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = 𝑤)
85 mulcomnqg 6843 . . . . . . . . . . . . . . 15 ((𝑤Q ∧ 1QQ) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8640, 85mpan2 416 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8784, 86eqtr3d 2117 . . . . . . . . . . . . 13 (𝑤Q𝑤 = (1Q ·Q 𝑤))
8887adantl 271 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → 𝑤 = (1Q ·Q 𝑤))
89 recidnq 6853 . . . . . . . . . . . . . 14 (𝑧Q → (𝑧 ·Q (*Q𝑧)) = 1Q)
9089oveq1d 5604 . . . . . . . . . . . . 13 (𝑧Q → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
9190adantr 270 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
92 mulassnqg 6844 . . . . . . . . . . . . . 14 ((𝑧Q ∧ (*Q𝑧) ∈ Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9331, 92syl3an2 1204 . . . . . . . . . . . . 13 ((𝑧Q𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
94933anidm12 1227 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9588, 91, 943eqtr2d 2121 . . . . . . . . . . 11 ((𝑧Q𝑤Q) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9683, 21, 95syl2anc 403 . . . . . . . . . 10 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
97 oveq2 5597 . . . . . . . . . . . 12 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9897eqeq2d 2094 . . . . . . . . . . 11 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))))
9998rspcev 2712 . . . . . . . . . 10 ((((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵) ∧ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))) → ∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥))
10082, 96, 99syl2anc 403 . . . . . . . . 9 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → ∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥))
1011003expia 1141 . . . . . . . 8 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (𝑣 <Q 𝑧 → ∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥)))
102101reximdv 2468 . . . . . . 7 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (∃𝑧 ∈ (1st𝐴)𝑣 <Q 𝑧 → ∃𝑧 ∈ (1st𝐴)∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥)))
10376recexprlempr 7092 . . . . . . . . 9 (𝐴P𝐵P)
104 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))}⟩)
105104, 55genpelvl 6972 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥)))
106103, 105mpdan 412 . . . . . . . 8 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥)))
107106ad2antrr 472 . . . . . . 7 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥)))
108102, 107sylibrd 167 . . . . . 6 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (∃𝑧 ∈ (1st𝐴)𝑣 <Q 𝑧𝑤 ∈ (1st ‘(𝐴 ·P 𝐵))))
10912, 108mpd 13 . . . . 5 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)))
1109, 109rexlimddv 2487 . . . 4 ((𝐴P𝑤 <Q 1Q) → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)))
111110ex 113 . . 3 (𝐴P → (𝑤 <Q 1Q𝑤 ∈ (1st ‘(𝐴 ·P 𝐵))))
1122, 111syl5bi 150 . 2 (𝐴P → (𝑤 ∈ (1st ‘1P) → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵))))
113112ssrdv 3016 1 (𝐴P → (1st ‘1P) ⊆ (1st ‘(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
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