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Theorem recexprlem1ssl 7409
Description: The lower cut of one is a subset of the lower cut of 𝐴 ·P 𝐵. Lemma for recexpr 7414. (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 7331 . . . 4 (1st ‘1P) = {𝑤𝑤 <Q 1Q}
21abeq2i 2228 . . 3 (𝑤 ∈ (1st ‘1P) ↔ 𝑤 <Q 1Q)
3 rec1nq 7171 . . . . . . 7 (*Q‘1Q) = 1Q
4 ltrnqi 7197 . . . . . . 7 (𝑤 <Q 1Q → (*Q‘1Q) <Q (*Q𝑤))
53, 4eqbrtrrid 3934 . . . . . 6 (𝑤 <Q 1Q → 1Q <Q (*Q𝑤))
6 prop 7251 . . . . . . 7 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
7 prmuloc2 7343 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ 1Q <Q (*Q𝑤)) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))
86, 7sylan 281 . . . . . 6 ((𝐴P ∧ 1Q <Q (*Q𝑤)) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))
95, 8sylan2 284 . . . . 5 ((𝐴P𝑤 <Q 1Q) → ∃𝑣 ∈ (1st𝐴)(𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))
10 prnmaxl 7264 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (1st𝐴)) → ∃𝑧 ∈ (1st𝐴)𝑣 <Q 𝑧)
116, 10sylan 281 . . . . . . 7 ((𝐴P𝑣 ∈ (1st𝐴)) → ∃𝑧 ∈ (1st𝐴)𝑣 <Q 𝑧)
1211ad2ant2r 500 . . . . . 6 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ∃𝑧 ∈ (1st𝐴)𝑣 <Q 𝑧)
13 elprnql 7257 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (1st𝐴)) → 𝑣Q)
146, 13sylan 281 . . . . . . . . . . . . 13 ((𝐴P𝑣 ∈ (1st𝐴)) → 𝑣Q)
1514ad2ant2r 500 . . . . . . . . . . . 12 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → 𝑣Q)
16153adant3 986 . . . . . . . . . . 11 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑣Q)
17 simp1r 991 . . . . . . . . . . . 12 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤 <Q 1Q)
18 ltrelnq 7141 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
1918brel 4561 . . . . . . . . . . . . 13 (𝑤 <Q 1Q → (𝑤Q ∧ 1QQ))
2019simpld 111 . . . . . . . . . . . 12 (𝑤 <Q 1Q𝑤Q)
2117, 20syl 14 . . . . . . . . . . 11 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤Q)
22 simp3 968 . . . . . . . . . . 11 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑣 <Q 𝑧)
23 simp2r 993 . . . . . . . . . . 11 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))
24 simpr 109 . . . . . . . . . . . 12 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)))
25 ltrnqi 7197 . . . . . . . . . . . . . 14 (𝑣 <Q 𝑧 → (*Q𝑧) <Q (*Q𝑣))
26 ltmnqg 7177 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( ·Q 𝑓) <Q ( ·Q 𝑔)))
2726adantl 275 . . . . . . . . . . . . . . 15 ((((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( ·Q 𝑓) <Q ( ·Q 𝑔)))
28 simprl 505 . . . . . . . . . . . . . . . 16 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → 𝑣 <Q 𝑧)
2918brel 4561 . . . . . . . . . . . . . . . . 17 (𝑣 <Q 𝑧 → (𝑣Q𝑧Q))
3029simprd 113 . . . . . . . . . . . . . . . 16 (𝑣 <Q 𝑧𝑧Q)
31 recclnq 7168 . . . . . . . . . . . . . . . 16 (𝑧Q → (*Q𝑧) ∈ Q)
3228, 30, 313syl 17 . . . . . . . . . . . . . . 15 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (*Q𝑧) ∈ Q)
33 recclnq 7168 . . . . . . . . . . . . . . . 16 (𝑣Q → (*Q𝑣) ∈ Q)
3433ad2antrr 479 . . . . . . . . . . . . . . 15 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (*Q𝑣) ∈ Q)
35 simplr 504 . . . . . . . . . . . . . . 15 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → 𝑤Q)
36 mulcomnqg 7159 . . . . . . . . . . . . . . . 16 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
3736adantl 275 . . . . . . . . . . . . . . 15 ((((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) ∧ (𝑓Q𝑔Q)) → (𝑓 ·Q 𝑔) = (𝑔 ·Q 𝑓))
3827, 32, 34, 35, 37caovord2d 5908 . . . . . . . . . . . . . 14 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ((*Q𝑧) <Q (*Q𝑣) ↔ ((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤)))
3925, 38syl5ib 153 . . . . . . . . . . . . 13 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (𝑣 <Q 𝑧 → ((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤)))
40 1nq 7142 . . . . . . . . . . . . . . . . . 18 1QQ
41 mulidnq 7165 . . . . . . . . . . . . . . . . . 18 (1QQ → (1Q ·Q 1Q) = 1Q)
4240, 41ax-mp 5 . . . . . . . . . . . . . . . . 17 (1Q ·Q 1Q) = 1Q
43 mulcomnqg 7159 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑣Q ∧ (*Q𝑣) ∈ Q) → (𝑣 ·Q (*Q𝑣)) = ((*Q𝑣) ·Q 𝑣))
4433, 43mpdan 417 . . . . . . . . . . . . . . . . . . . . 21 (𝑣Q → (𝑣 ·Q (*Q𝑣)) = ((*Q𝑣) ·Q 𝑣))
45 recidnq 7169 . . . . . . . . . . . . . . . . . . . . 21 (𝑣Q → (𝑣 ·Q (*Q𝑣)) = 1Q)
4644, 45eqtr3d 2152 . . . . . . . . . . . . . . . . . . . 20 (𝑣Q → ((*Q𝑣) ·Q 𝑣) = 1Q)
47 recidnq 7169 . . . . . . . . . . . . . . . . . . . 20 (𝑤Q → (𝑤 ·Q (*Q𝑤)) = 1Q)
4846, 47oveqan12d 5761 . . . . . . . . . . . . . . . . . . 19 ((𝑣Q𝑤Q) → (((*Q𝑣) ·Q 𝑣) ·Q (𝑤 ·Q (*Q𝑤))) = (1Q ·Q 1Q))
4948adantr 274 . . . . . . . . . . . . . . . . . 18 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (((*Q𝑣) ·Q 𝑣) ·Q (𝑤 ·Q (*Q𝑤))) = (1Q ·Q 1Q))
50 simpll 503 . . . . . . . . . . . . . . . . . . 19 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → 𝑣Q)
51 mulassnqg 7160 . . . . . . . . . . . . . . . . . . . 20 ((𝑓Q𝑔QQ) → ((𝑓 ·Q 𝑔) ·Q ) = (𝑓 ·Q (𝑔 ·Q )))
5251adantl 275 . . . . . . . . . . . . . . . . . . 19 ((((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) ∧ (𝑓Q𝑔QQ)) → ((𝑓 ·Q 𝑔) ·Q ) = (𝑓 ·Q (𝑔 ·Q )))
53 recclnq 7168 . . . . . . . . . . . . . . . . . . . 20 (𝑤Q → (*Q𝑤) ∈ Q)
5435, 53syl 14 . . . . . . . . . . . . . . . . . . 19 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (*Q𝑤) ∈ Q)
55 mulclnq 7152 . . . . . . . . . . . . . . . . . . . 20 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
5655adantl 275 . . . . . . . . . . . . . . . . . . 19 ((((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) ∧ (𝑓Q𝑔Q)) → (𝑓 ·Q 𝑔) ∈ Q)
5734, 50, 35, 37, 52, 54, 56caov4d 5923 . . . . . . . . . . . . . . . . . 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 2152 . . . . . . . . . . . . . . . . 17 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (1Q ·Q 1Q) = (((*Q𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q𝑤))))
5942, 58syl5reqr 2165 . . . . . . . . . . . . . . . 16 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (((*Q𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q𝑤))) = 1Q)
60 mulclnq 7152 . . . . . . . . . . . . . . . . . . 19 (((*Q𝑣) ∈ Q𝑤Q) → ((*Q𝑣) ·Q 𝑤) ∈ Q)
6133, 60sylan 281 . . . . . . . . . . . . . . . . . 18 ((𝑣Q𝑤Q) → ((*Q𝑣) ·Q 𝑤) ∈ Q)
62 mulclnq 7152 . . . . . . . . . . . . . . . . . . 19 ((𝑣Q ∧ (*Q𝑤) ∈ Q) → (𝑣 ·Q (*Q𝑤)) ∈ Q)
6353, 62sylan2 284 . . . . . . . . . . . . . . . . . 18 ((𝑣Q𝑤Q) → (𝑣 ·Q (*Q𝑤)) ∈ Q)
64 recmulnqg 7167 . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . 17 ((𝑣Q𝑤Q) → ((*Q‘((*Q𝑣) ·Q 𝑤)) = (𝑣 ·Q (*Q𝑤)) ↔ (((*Q𝑣) ·Q 𝑤) ·Q (𝑣 ·Q (*Q𝑤))) = 1Q))
6665adantr 274 . . . . . . . . . . . . . . . 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 166 . . . . . . . . . . . . . . 15 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (*Q‘((*Q𝑣) ·Q 𝑤)) = (𝑣 ·Q (*Q𝑤)))
6867eleq1d 2186 . . . . . . . . . . . . . 14 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ((*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴) ↔ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)))
6968biimprd 157 . . . . . . . . . . . . 13 (((𝑣Q𝑤Q) ∧ (𝑣 <Q 𝑧 ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → ((𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴) → (*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴)))
70 breq2 3903 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((*Q𝑣) ·Q 𝑤) → (((*Q𝑧) ·Q 𝑤) <Q 𝑦 ↔ ((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤)))
71 fveq2 5389 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((*Q𝑣) ·Q 𝑤) → (*Q𝑦) = (*Q‘((*Q𝑣) ·Q 𝑤)))
7271eleq1d 2186 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((*Q𝑣) ·Q 𝑤) → ((*Q𝑦) ∈ (2nd𝐴) ↔ (*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴)))
7370, 72anbi12d 464 . . . . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . . . . . 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 7398 . . . . . . . . . . . . . . 15 (((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵) ↔ ∃𝑦(((*Q𝑧) ·Q 𝑤) <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
7875, 77syl6ibr 161 . . . . . . . . . . . . . 14 ((𝑣Q𝑤Q) → ((((*Q𝑧) ·Q 𝑤) <Q ((*Q𝑣) ·Q 𝑤) ∧ (*Q‘((*Q𝑣) ·Q 𝑤)) ∈ (2nd𝐴)) → ((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵)))
7978adantr 274 . . . . . . . . . . . . 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 293 . . . . . . . . . . . 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 1202 . . . . . . . . . 10 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → ((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵))
83303ad2ant3 989 . . . . . . . . . . 11 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑧Q)
84 mulidnq 7165 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = 𝑤)
85 mulcomnqg 7159 . . . . . . . . . . . . . . 15 ((𝑤Q ∧ 1QQ) → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8640, 85mpan2 421 . . . . . . . . . . . . . 14 (𝑤Q → (𝑤 ·Q 1Q) = (1Q ·Q 𝑤))
8784, 86eqtr3d 2152 . . . . . . . . . . . . 13 (𝑤Q𝑤 = (1Q ·Q 𝑤))
8887adantl 275 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → 𝑤 = (1Q ·Q 𝑤))
89 recidnq 7169 . . . . . . . . . . . . . 14 (𝑧Q → (𝑧 ·Q (*Q𝑧)) = 1Q)
9089oveq1d 5757 . . . . . . . . . . . . 13 (𝑧Q → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
9190adantr 274 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (1Q ·Q 𝑤))
92 mulassnqg 7160 . . . . . . . . . . . . . 14 ((𝑧Q ∧ (*Q𝑧) ∈ Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9331, 92syl3an2 1235 . . . . . . . . . . . . 13 ((𝑧Q𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
94933anidm12 1258 . . . . . . . . . . . 12 ((𝑧Q𝑤Q) → ((𝑧 ·Q (*Q𝑧)) ·Q 𝑤) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9588, 91, 943eqtr2d 2156 . . . . . . . . . . 11 ((𝑧Q𝑤Q) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9683, 21, 95syl2anc 408 . . . . . . . . . 10 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
97 oveq2 5750 . . . . . . . . . . . 12 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑧 ·Q 𝑥) = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤)))
9897eqeq2d 2129 . . . . . . . . . . 11 (𝑥 = ((*Q𝑧) ·Q 𝑤) → (𝑤 = (𝑧 ·Q 𝑥) ↔ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))))
9998rspcev 2763 . . . . . . . . . 10 ((((*Q𝑧) ·Q 𝑤) ∈ (1st𝐵) ∧ 𝑤 = (𝑧 ·Q ((*Q𝑧) ·Q 𝑤))) → ∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥))
10082, 96, 99syl2anc 408 . . . . . . . . 9 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴)) ∧ 𝑣 <Q 𝑧) → ∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥))
1011003expia 1168 . . . . . . . 8 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (𝑣 <Q 𝑧 → ∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥)))
102101reximdv 2510 . . . . . . 7 (((𝐴P𝑤 <Q 1Q) ∧ (𝑣 ∈ (1st𝐴) ∧ (𝑣 ·Q (*Q𝑤)) ∈ (2nd𝐴))) → (∃𝑧 ∈ (1st𝐴)𝑣 <Q 𝑧 → ∃𝑧 ∈ (1st𝐴)∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥)))
10376recexprlempr 7408 . . . . . . . . 9 (𝐴P𝐵P)
104 df-imp 7245 . . . . . . . . . 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 7288 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥)))
106103, 105mpdan 417 . . . . . . . 8 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑥 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑥)))
107106ad2antrr 479 . . . . . . 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 168 . . . . . 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 2531 . . . 4 ((𝐴P𝑤 <Q 1Q) → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)))
111110ex 114 . . 3 (𝐴P → (𝑤 <Q 1Q𝑤 ∈ (1st ‘(𝐴 ·P 𝐵))))
1122, 111syl5bi 151 . 2 (𝐴P → (𝑤 ∈ (1st ‘1P) → 𝑤 ∈ (1st ‘(𝐴 ·P 𝐵))))
113112ssrdv 3073 1 (𝐴P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 947   = wceq 1316  wex 1453  wcel 1465  {cab 2103  wrex 2394  wss 3041  cop 3500   class class class wbr 3899  cfv 5093  (class class class)co 5742  1st c1st 6004  2nd c2nd 6005  Qcnq 7056  1Qc1q 7057   ·Q cmq 7059  *Qcrq 7060   <Q cltq 7061  Pcnp 7067  1Pc1p 7068   ·P cmp 7070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-i1p 7243  df-imp 7245
This theorem is referenced by:  recexprlemex  7413
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