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Theorem mulextsr1 7779
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 7725 . 2 R = ((P × P) / ~R )
2 oveq1 5881 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ))
32breq1d 4013 . . 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 4006 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R [⟨𝑧, 𝑤⟩] ~R ))
5 breq2 4007 . . . 4 ([⟨𝑥, 𝑦⟩] ~R = 𝐴 → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ [⟨𝑧, 𝑤⟩] ~R <R 𝐴))
64, 5orbi12d 793 . . 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 234 . 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 5881 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ))
98breq2d 4015 . . 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 4007 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → (𝐴 <R [⟨𝑧, 𝑤⟩] ~R𝐴 <R 𝐵))
11 breq1 4006 . . . 4 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ([⟨𝑧, 𝑤⟩] ~R <R 𝐴𝐵 <R 𝐴))
1210, 11orbi12d 793 . . 3 ([⟨𝑧, 𝑤⟩] ~R = 𝐵 → ((𝐴 <R [⟨𝑧, 𝑤⟩] ~R ∨ [⟨𝑧, 𝑤⟩] ~R <R 𝐴) ↔ (𝐴 <R 𝐵𝐵 <R 𝐴)))
139, 12imbi12d 234 . 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 5882 . . . 4 ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐴 ·R 𝐶))
15 oveq2 5882 . . . 4 ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) = (𝐵 ·R 𝐶))
1614, 15breq12d 4016 . . 3 ([⟨𝑢, 𝑣⟩] ~R = 𝐶 → ((𝐴 ·R [⟨𝑢, 𝑣⟩] ~R ) <R (𝐵 ·R [⟨𝑢, 𝑣⟩] ~R ) ↔ (𝐴 ·R 𝐶) <R (𝐵 ·R 𝐶)))
1716imbi1d 231 . 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 7778 . . 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 7744 . . . . . 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 1016 . . . . 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 7744 . . . . . 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 1015 . . . . 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 4016 . . . 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 1021 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑥P)
25 simp3l 1025 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑢P)
26 mulclpr 7570 . . . . . . 7 ((𝑥P𝑢P) → (𝑥 ·P 𝑢) ∈ P)
2724, 25, 26syl2anc 411 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑥 ·P 𝑢) ∈ P)
28 simp1r 1022 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑦P)
29 simp3r 1026 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑣P)
30 mulclpr 7570 . . . . . . 7 ((𝑦P𝑣P) → (𝑦 ·P 𝑣) ∈ P)
3128, 29, 30syl2anc 411 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑦 ·P 𝑣) ∈ P)
32 addclpr 7535 . . . . . 6 (((𝑥 ·P 𝑢) ∈ P ∧ (𝑦 ·P 𝑣) ∈ P) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
3327, 31, 32syl2anc 411 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ((𝑥 ·P 𝑢) +P (𝑦 ·P 𝑣)) ∈ P)
34 mulclpr 7570 . . . . . . 7 ((𝑥P𝑣P) → (𝑥 ·P 𝑣) ∈ P)
3524, 29, 34syl2anc 411 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑥 ·P 𝑣) ∈ P)
36 mulclpr 7570 . . . . . . 7 ((𝑦P𝑢P) → (𝑦 ·P 𝑢) ∈ P)
3728, 25, 36syl2anc 411 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑦 ·P 𝑢) ∈ P)
38 addclpr 7535 . . . . . 6 (((𝑥 ·P 𝑣) ∈ P ∧ (𝑦 ·P 𝑢) ∈ P) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
3935, 37, 38syl2anc 411 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ((𝑥 ·P 𝑣) +P (𝑦 ·P 𝑢)) ∈ P)
40 simp2l 1023 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑧P)
41 mulclpr 7570 . . . . . . 7 ((𝑧P𝑢P) → (𝑧 ·P 𝑢) ∈ P)
4240, 25, 41syl2anc 411 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑧 ·P 𝑢) ∈ P)
43 simp2r 1024 . . . . . . 7 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → 𝑤P)
44 mulclpr 7570 . . . . . . 7 ((𝑤P𝑣P) → (𝑤 ·P 𝑣) ∈ P)
4543, 29, 44syl2anc 411 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑤 ·P 𝑣) ∈ P)
46 addclpr 7535 . . . . . 6 (((𝑧 ·P 𝑢) ∈ P ∧ (𝑤 ·P 𝑣) ∈ P) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
4742, 45, 46syl2anc 411 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ((𝑧 ·P 𝑢) +P (𝑤 ·P 𝑣)) ∈ P)
48 mulclpr 7570 . . . . . . 7 ((𝑧P𝑣P) → (𝑧 ·P 𝑣) ∈ P)
4940, 29, 48syl2anc 411 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑧 ·P 𝑣) ∈ P)
50 mulclpr 7570 . . . . . . 7 ((𝑤P𝑢P) → (𝑤 ·P 𝑢) ∈ P)
5143, 25, 50syl2anc 411 . . . . . 6 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → (𝑤 ·P 𝑢) ∈ P)
52 addclpr 7535 . . . . . 6 (((𝑧 ·P 𝑣) ∈ P ∧ (𝑤 ·P 𝑢) ∈ P) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
5349, 51, 52syl2anc 411 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑢)) ∈ P)
54 ltsrprg 7745 . . . . 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 1239 . . . 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 188 . . 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 7745 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
58573adant3 1017 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ([⟨𝑥, 𝑦⟩] ~R <R [⟨𝑧, 𝑤⟩] ~R ↔ (𝑥 +P 𝑤)<P (𝑦 +P 𝑧)))
59 ltsrprg 7745 . . . . . 6 (((𝑧P𝑤P) ∧ (𝑥P𝑦P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
6059ancoms 268 . . . . 5 (((𝑥P𝑦P) ∧ (𝑧P𝑤P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
61603adant3 1017 . . . 4 (((𝑥P𝑦P) ∧ (𝑧P𝑤P) ∧ (𝑢P𝑣P)) → ([⟨𝑧, 𝑤⟩] ~R <R [⟨𝑥, 𝑦⟩] ~R ↔ (𝑧 +P 𝑦)<P (𝑤 +P 𝑥)))
6258, 61orbi12d 793 . . 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 203 . 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 6623 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 104  wb 105  wo 708  w3a 978   = wceq 1353  wcel 2148  cop 3595   class class class wbr 4003  (class class class)co 5874  [cec 6532  Pcnp 7289   +P cpp 7291   ·P cmp 7292  <P cltp 7293   ~R cer 7294  Rcnr 7295   ·R cmr 7300   <R cltr 7301
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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-i1p 7465  df-iplp 7466  df-imp 7467  df-iltp 7468  df-enr 7724  df-nr 7725  df-mr 7727  df-ltr 7728
This theorem is referenced by:  axpre-mulext  7886
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