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Theorem mulcmpblnrlem 9747
Description: Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulcmpblnrlem (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))))

Proof of Theorem mulcmpblnrlem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6534 . . . . . . . . 9 ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐵 +P 𝐶) ·P 𝐹))
2 distrpr 9706 . . . . . . . . . 10 (𝐹 ·P (𝐴 +P 𝐷)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷))
3 mulcompr 9701 . . . . . . . . . 10 ((𝐴 +P 𝐷) ·P 𝐹) = (𝐹 ·P (𝐴 +P 𝐷))
4 mulcompr 9701 . . . . . . . . . . 11 (𝐴 ·P 𝐹) = (𝐹 ·P 𝐴)
5 mulcompr 9701 . . . . . . . . . . 11 (𝐷 ·P 𝐹) = (𝐹 ·P 𝐷)
64, 5oveq12i 6539 . . . . . . . . . 10 ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) = ((𝐹 ·P 𝐴) +P (𝐹 ·P 𝐷))
72, 3, 63eqtr4i 2641 . . . . . . . . 9 ((𝐴 +P 𝐷) ·P 𝐹) = ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹))
8 distrpr 9706 . . . . . . . . . 10 (𝐹 ·P (𝐵 +P 𝐶)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶))
9 mulcompr 9701 . . . . . . . . . 10 ((𝐵 +P 𝐶) ·P 𝐹) = (𝐹 ·P (𝐵 +P 𝐶))
10 mulcompr 9701 . . . . . . . . . . 11 (𝐵 ·P 𝐹) = (𝐹 ·P 𝐵)
11 mulcompr 9701 . . . . . . . . . . 11 (𝐶 ·P 𝐹) = (𝐹 ·P 𝐶)
1210, 11oveq12i 6539 . . . . . . . . . 10 ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) = ((𝐹 ·P 𝐵) +P (𝐹 ·P 𝐶))
138, 9, 123eqtr4i 2641 . . . . . . . . 9 ((𝐵 +P 𝐶) ·P 𝐹) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹))
141, 7, 133eqtr3g 2666 . . . . . . . 8 ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) = ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)))
1514oveq1d 6542 . . . . . . 7 ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)))
16 addasspr 9700 . . . . . . . 8 (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆)))
17 oveq2 6535 . . . . . . . . . 10 ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (𝐶 ·P (𝐹 +P 𝑆)) = (𝐶 ·P (𝐺 +P 𝑅)))
18 distrpr 9706 . . . . . . . . . 10 (𝐶 ·P (𝐹 +P 𝑆)) = ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆))
19 distrpr 9706 . . . . . . . . . 10 (𝐶 ·P (𝐺 +P 𝑅)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))
2017, 18, 193eqtr3g 2666 . . . . . . . . 9 ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆)) = ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅)))
2120oveq2d 6543 . . . . . . . 8 ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐹) +P (𝐶 ·P 𝑆))) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))))
2216, 21syl5eq 2655 . . . . . . 7 ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (((𝐵 ·P 𝐹) +P (𝐶 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))))
2315, 22sylan9eq 2663 . . . . . 6 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))))
24 ovex 6555 . . . . . . 7 (𝐴 ·P 𝐹) ∈ V
25 ovex 6555 . . . . . . 7 (𝐷 ·P 𝐹) ∈ V
26 ovex 6555 . . . . . . 7 (𝐶 ·P 𝑆) ∈ V
27 addcompr 9699 . . . . . . 7 (𝑥 +P 𝑦) = (𝑦 +P 𝑥)
28 addasspr 9700 . . . . . . 7 ((𝑥 +P 𝑦) +P 𝑧) = (𝑥 +P (𝑦 +P 𝑧))
2924, 25, 26, 27, 28caov32 6736 . . . . . 6 (((𝐴 ·P 𝐹) +P (𝐷 ·P 𝐹)) +P (𝐶 ·P 𝑆)) = (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))
30 ovex 6555 . . . . . . 7 (𝐵 ·P 𝐹) ∈ V
31 ovex 6555 . . . . . . 7 (𝐶 ·P 𝐺) ∈ V
32 ovex 6555 . . . . . . 7 (𝐶 ·P 𝑅) ∈ V
3330, 31, 32, 27, 28caov12 6737 . . . . . 6 ((𝐵 ·P 𝐹) +P ((𝐶 ·P 𝐺) +P (𝐶 ·P 𝑅))) = ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))
3423, 29, 333eqtr3g 2666 . . . . 5 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹)) = ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
3534oveq2d 6543 . . . 4 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
36 oveq2 6535 . . . . . . . . . . 11 ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → (𝐷 ·P (𝐹 +P 𝑆)) = (𝐷 ·P (𝐺 +P 𝑅)))
37 distrpr 9706 . . . . . . . . . . 11 (𝐷 ·P (𝐹 +P 𝑆)) = ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))
38 distrpr 9706 . . . . . . . . . . 11 (𝐷 ·P (𝐺 +P 𝑅)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅))
3936, 37, 383eqtr3g 2666 . . . . . . . . . 10 ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆)) = ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅)))
4039oveq2d 6543 . . . . . . . . 9 ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅))))
41 addasspr 9700 . . . . . . . . 9 (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐺) +P (𝐷 ·P 𝑅)))
4240, 41syl6eqr 2661 . . . . . . . 8 ((𝐹 +P 𝑆) = (𝐺 +P 𝑅) → ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) +P (𝐷 ·P 𝑅)))
43 oveq1 6534 . . . . . . . . . 10 ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐵 +P 𝐶) ·P 𝐺))
44 distrpr 9706 . . . . . . . . . . 11 (𝐺 ·P (𝐴 +P 𝐷)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷))
45 mulcompr 9701 . . . . . . . . . . 11 ((𝐴 +P 𝐷) ·P 𝐺) = (𝐺 ·P (𝐴 +P 𝐷))
46 mulcompr 9701 . . . . . . . . . . . 12 (𝐴 ·P 𝐺) = (𝐺 ·P 𝐴)
47 mulcompr 9701 . . . . . . . . . . . 12 (𝐷 ·P 𝐺) = (𝐺 ·P 𝐷)
4846, 47oveq12i 6539 . . . . . . . . . . 11 ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) = ((𝐺 ·P 𝐴) +P (𝐺 ·P 𝐷))
4944, 45, 483eqtr4i 2641 . . . . . . . . . 10 ((𝐴 +P 𝐷) ·P 𝐺) = ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺))
50 distrpr 9706 . . . . . . . . . . 11 (𝐺 ·P (𝐵 +P 𝐶)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶))
51 mulcompr 9701 . . . . . . . . . . 11 ((𝐵 +P 𝐶) ·P 𝐺) = (𝐺 ·P (𝐵 +P 𝐶))
52 mulcompr 9701 . . . . . . . . . . . 12 (𝐵 ·P 𝐺) = (𝐺 ·P 𝐵)
53 mulcompr 9701 . . . . . . . . . . . 12 (𝐶 ·P 𝐺) = (𝐺 ·P 𝐶)
5452, 53oveq12i 6539 . . . . . . . . . . 11 ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) = ((𝐺 ·P 𝐵) +P (𝐺 ·P 𝐶))
5550, 51, 543eqtr4i 2641 . . . . . . . . . 10 ((𝐵 +P 𝐶) ·P 𝐺) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺))
5643, 49, 553eqtr3g 2666 . . . . . . . . 9 ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) = ((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)))
5756oveq1d 6542 . . . . . . . 8 ((𝐴 +P 𝐷) = (𝐵 +P 𝐶) → (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = (((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) +P (𝐷 ·P 𝑅)))
5842, 57sylan9eqr 2665 . . . . . . 7 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = (((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) +P (𝐷 ·P 𝑅)))
59 ovex 6555 . . . . . . . 8 (𝐴 ·P 𝐺) ∈ V
60 ovex 6555 . . . . . . . 8 (𝐷 ·P 𝑆) ∈ V
6159, 25, 60, 27, 28caov12 6737 . . . . . . 7 ((𝐴 ·P 𝐺) +P ((𝐷 ·P 𝐹) +P (𝐷 ·P 𝑆))) = ((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)))
62 ovex 6555 . . . . . . . 8 (𝐵 ·P 𝐺) ∈ V
63 ovex 6555 . . . . . . . 8 (𝐷 ·P 𝑅) ∈ V
6462, 31, 63, 27, 28caov32 6736 . . . . . . 7 (((𝐵 ·P 𝐺) +P (𝐶 ·P 𝐺)) +P (𝐷 ·P 𝑅)) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺))
6558, 61, 643eqtr3g 2666 . . . . . 6 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)))
6665oveq1d 6542 . . . . 5 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = ((((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
67 addasspr 9700 . . . . 5 ((((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (𝐶 ·P 𝐺)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
6866, 67syl6eq 2659 . . . 4 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P ((𝐶 ·P 𝐺) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
6935, 68eqtr4d 2646 . . 3 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))) = (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
70 ovex 6555 . . . 4 ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) ∈ V
71 ovex 6555 . . . 4 ((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) ∈ V
7270, 71, 25, 27, 28caov13 6739 . . 3 (((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P (𝐷 ·P 𝐹))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅))))
73 addasspr 9700 . . 3 (((𝐷 ·P 𝐹) +P ((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆))) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))))
7469, 72, 733eqtr3g 2666 . 2 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))))
7524, 26, 62, 27, 28, 63caov4 6740 . . 3 (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅))) = (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))
7675oveq2i 6538 . 2 ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐶 ·P 𝑆)) +P ((𝐵 ·P 𝐺) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅))))
7759, 60, 30, 27, 28, 32caov42 6742 . . 3 (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅))) = (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))
7877oveq2i 6538 . 2 ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐷 ·P 𝑆)) +P ((𝐵 ·P 𝐹) +P (𝐶 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆))))
7974, 76, 783eqtr3g 2666 1 (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  (class class class)co 6527   +P cpp 9539   ·P cmp 9540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-ni 9550  df-pli 9551  df-mi 9552  df-lti 9553  df-plpq 9586  df-mpq 9587  df-ltpq 9588  df-enq 9589  df-nq 9590  df-erq 9591  df-plq 9592  df-mq 9593  df-1nq 9594  df-rq 9595  df-ltnq 9596  df-np 9659  df-plp 9661  df-mp 9662
This theorem is referenced by:  mulcmpblnr  9748
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