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Theorem mulextsr1 7270
Description: Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)
Assertion
Ref Expression
mulextsr1 ((𝐴R𝐵R𝐶R) → ((𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶) → (𝐴 <R 𝐵𝐵 <R 𝐴)))

Proof of Theorem mulextsr1
Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7217 . 2 R = ((P × P) / ~R )
2 oveq1 5620 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ))
32breq1d 3830 . . 3 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R )))
4 breq1 3823 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
5 breq2 3824 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ [⟨𝑧, 𝑤⟩] ~R <R 𝐴))
64, 5orbi12d 740 . . 3 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴)))
73, 6imbi12d 232 . 2 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ((([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R )) ↔ ((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴))))
8 oveq1 5620 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ))
98breq2d 3832 . . 3 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R )))
10 breq2 3824 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R 𝐵))
11 breq1 3823 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R <R 𝐴𝐵 <R 𝐴))
1210, 11orbi12d 740 . . 3 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) ↔ (𝐴 <R 𝐵𝐵 <R 𝐴)))
139, 12imbi12d 232 . 2 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴)) ↔ ((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) → (𝐴 <R 𝐵𝐵 <R 𝐴))))
14 oveq2 5621 . . . 4 ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐴 ·R 𝐶))
15 oveq2 5621 . . . 4 ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐵 ·R 𝐶))
1614, 15breq12d 3833 . . 3 ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → ((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ (𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶)))
1716imbi1d 229 . 2 ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) → (𝐴 <R 𝐵𝐵 <R 𝐴)) ↔ ((𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶) → (𝐴 <R 𝐵𝐵 <R 𝐴))))
18 mulextsr1lem 7269 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ((((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) +P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)))<P (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) +P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣))) → ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))))
19 mulsrpr 7236 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑢P𝑣P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = [⟨((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)), ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))⟩] ~R )
20193adant2 960 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = [⟨((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)), ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))⟩] ~R )
21 mulsrpr 7236 . . . . . 6 (((𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = [⟨((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)), ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))⟩] ~R )
22213adant1 959 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = [⟨((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)), ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))⟩] ~R )
2320, 22breq12d 3833 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ [⟨((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)), ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))⟩] ~R <R [⟨((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)), ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))⟩] ~R ))
24 simp1l 965 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑥P)
25 simp3l 969 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑢P)
26 mulclpr 7075 . . . . . . 7 ((𝑥P𝑢P) → (𝑥 ·P 𝑢) ∈ P)
2724, 25, 26syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑥 ·P 𝑢) ∈ P)
28 simp1r 966 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑦P)
29 simp3r 970 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑣P)
30 mulclpr 7075 . . . . . . 7 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
3128, 29, 30syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑦 ·P 𝑣) ∈ P)
32 addclpr 7040 . . . . . 6 (((𝑥 ·P 𝑢) ∈ P ∧ (𝑦 ·P 𝑣) ∈ P) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3327, 31, 32syl2anc 403 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
34 mulclpr 7075 . . . . . . 7 ((𝑥P𝑣P) → (𝑥 ·P 𝑣) ∈ P)
3524, 29, 34syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑥 ·P 𝑣) ∈ P)
36 mulclpr 7075 . . . . . . 7 ((𝑦P𝑢P) → (𝑦 ·P 𝑢) ∈ P)
3728, 25, 36syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑦 ·P 𝑢) ∈ P)
38 addclpr 7040 . . . . . 6 (((𝑥 ·P 𝑣) ∈ P ∧ (𝑦 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
3935, 37, 38syl2anc 403 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
40 simp2l 967 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑧P)
41 mulclpr 7075 . . . . . . 7 ((𝑧P𝑢P) → (𝑧 ·P 𝑢) ∈ P)
4240, 25, 41syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑧 ·P 𝑢) ∈ P)
43 simp2r 968 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑤P)
44 mulclpr 7075 . . . . . . 7 ((𝑤P𝑣P) → (𝑤 ·P 𝑣) ∈ P)
4543, 29, 44syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑤 ·P 𝑣) ∈ P)
46 addclpr 7040 . . . . . 6 (((𝑧 ·P 𝑢) ∈ P ∧ (𝑤 ·P 𝑣) ∈ P) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
4742, 45, 46syl2anc 403 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
48 mulclpr 7075 . . . . . . 7 ((𝑧P𝑣P) → (𝑧 ·P 𝑣) ∈ P)
4940, 29, 48syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑧 ·P 𝑣) ∈ P)
50 mulclpr 7075 . . . . . . 7 ((𝑤P𝑢P) → (𝑤 ·P 𝑢) ∈ P)
5143, 25, 50syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑤 ·P 𝑢) ∈ P)
52 addclpr 7040 . . . . . 6 (((𝑧 ·P 𝑣) ∈ P ∧ (𝑤 ·P 𝑢) ∈ P) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
5349, 51, 52syl2anc 403 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
54 ltsrprg 7237 . . . . 5 (((((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P ∧ ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P) ∧ (((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P ∧ ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)) → ([⟨((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)), ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))⟩] ~R <R [⟨((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)), ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))⟩] ~R ↔ (((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) +P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)))<P (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) +P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)))))
5533, 39, 47, 53, 54syl22anc 1173 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ([⟨((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)), ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢))⟩] ~R <R [⟨((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)), ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢))⟩] ~R ↔ (((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) +P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)))<P (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) +P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)))))
5623, 55bitrd 186 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ (((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) +P ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)))<P (((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) +P ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)))))
57 ltsrprg 7237 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
58573adant3 961 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
59 ltsrprg 7237 . . . . . 6 (((𝑧P𝑤P) ∧ (𝑥P𝑦P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
6059ancoms 264 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
61603adant3 961 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
6258, 61orbi12d 740 . . 3 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ) ↔ ((𝑥 +P 𝑤)<P (𝑦 +P 𝑧) ∨ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥))))
6318, 56, 623imtr4d 201 . 2 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) <R ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R )))
641, 7, 13, 17, 633ecoptocl 6333 1 ((𝐴R𝐵R𝐶R) → ((𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶) → (𝐴 <R 𝐵𝐵 <R 𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  w3a 922   = wceq 1287  wcel 1436  cop 3434   class class class wbr 3820  (class class class)co 5613  [cec 6242  Pcnp 6794   +P cpp 6796   ·P cmp 6797  <P cltp 6798   ~R cer 6799  Rcnr 6800   ·R cmr 6805   <R cltr 6806
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-eprel 4090  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-1o 6135  df-2o 6136  df-oadd 6139  df-omul 6140  df-er 6244  df-ec 6246  df-qs 6250  df-ni 6807  df-pli 6808  df-mi 6809  df-lti 6810  df-plpq 6847  df-mpq 6848  df-enq 6850  df-nqqs 6851  df-plqqs 6852  df-mqqs 6853  df-1nqqs 6854  df-rq 6855  df-ltnqqs 6856  df-enq0 6927  df-nq0 6928  df-0nq0 6929  df-plq0 6930  df-mq0 6931  df-inp 6969  df-i1p 6970  df-iplp 6971  df-imp 6972  df-iltp 6973  df-enr 7216  df-nr 7217  df-mr 7219  df-ltr 7220
This theorem is referenced by:  axpre-mulext  7367
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