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Theorem mulextsr1 7019
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 6966 . 2 R = ((P × P) / ~R )
2 oveq1 5550 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ))
32breq1d 3803 . . 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 3796 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
5 breq2 3797 . . . 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 5550 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ))
98breq2d 3805 . . 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 3797 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R 𝐵))
11 breq1 3796 . . . 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 5551 . . . 4 ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐴 ·R 𝐶))
15 oveq2 5551 . . . 4 ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐵 ·R 𝐶))
1614, 15breq12d 3806 . . 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 7018 . . 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 6985 . . . . . 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 958 . . . . 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 6985 . . . . . 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 957 . . . . 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 3806 . . . 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 963 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑥P)
25 simp3l 967 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑢P)
26 mulclpr 6824 . . . . . . 7 ((𝑥P𝑢P) → (𝑥 ·P 𝑢) ∈ P)
2724, 25, 26syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑥 ·P 𝑢) ∈ P)
28 simp1r 964 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑦P)
29 simp3r 968 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑣P)
30 mulclpr 6824 . . . . . . 7 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
3128, 29, 30syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑦 ·P 𝑣) ∈ P)
32 addclpr 6789 . . . . . 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 6824 . . . . . . 7 ((𝑥P𝑣P) → (𝑥 ·P 𝑣) ∈ P)
3524, 29, 34syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑥 ·P 𝑣) ∈ P)
36 mulclpr 6824 . . . . . . 7 ((𝑦P𝑢P) → (𝑦 ·P 𝑢) ∈ P)
3728, 25, 36syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑦 ·P 𝑢) ∈ P)
38 addclpr 6789 . . . . . 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 965 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑧P)
41 mulclpr 6824 . . . . . . 7 ((𝑧P𝑢P) → (𝑧 ·P 𝑢) ∈ P)
4240, 25, 41syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑧 ·P 𝑢) ∈ P)
43 simp2r 966 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑤P)
44 mulclpr 6824 . . . . . . 7 ((𝑤P𝑣P) → (𝑤 ·P 𝑣) ∈ P)
4543, 29, 44syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑤 ·P 𝑣) ∈ P)
46 addclpr 6789 . . . . . 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 6824 . . . . . . 7 ((𝑧P𝑣P) → (𝑧 ·P 𝑣) ∈ P)
4940, 29, 48syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑧 ·P 𝑣) ∈ P)
50 mulclpr 6824 . . . . . . 7 ((𝑤P𝑢P) → (𝑤 ·P 𝑢) ∈ P)
5143, 25, 50syl2anc 403 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑤 ·P 𝑢) ∈ P)
52 addclpr 6789 . . . . . 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 6986 . . . . 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 1171 . . . 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 6986 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
58573adant3 959 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
59 ltsrprg 6986 . . . . . 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 959 . . . 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 6261 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 920   = wceq 1285  wcel 1434  cop 3409   class class class wbr 3793  (class class class)co 5543  [cec 6170  Pcnp 6543   +P cpp 6545   ·P cmp 6546  <P cltp 6547   ~R cer 6548  Rcnr 6549   ·R cmr 6554   <R cltr 6555
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 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-i1p 6719  df-iplp 6720  df-imp 6721  df-iltp 6722  df-enr 6965  df-nr 6966  df-mr 6968  df-ltr 6969
This theorem is referenced by:  axpre-mulext  7116
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