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Axiom ax-pre-mulext 6761
Description: Strong extensionality of multiplication (expressed in terms of <). Axiom for real and complex numbers, justified by theorem axpre-mulext 6732

(Contributed by Jim Kingdon, 18-Feb-2020.)

Assertion
Ref Expression
ax-pre-mulext ((A B 𝐶 ℝ) → ((A · 𝐶) < (B · 𝐶) → (A < B B < A)))

Detailed syntax breakdown of Axiom ax-pre-mulext
StepHypRef Expression
1 cA . . . 4 class A
2 cr 6670 . . . 4 class
31, 2wcel 1390 . . 3 wff A
4 cB . . . 4 class B
54, 2wcel 1390 . . 3 wff B
6 cC . . . 4 class 𝐶
76, 2wcel 1390 . . 3 wff 𝐶
83, 5, 7w3a 884 . 2 wff (A B 𝐶 ℝ)
9 cmul 6676 . . . . 5 class ·
101, 6, 9co 5455 . . . 4 class (A · 𝐶)
114, 6, 9co 5455 . . . 4 class (B · 𝐶)
12 cltrr 6675 . . . 4 class <
1310, 11, 12wbr 3755 . . 3 wff (A · 𝐶) < (B · 𝐶)
141, 4, 12wbr 3755 . . . 4 wff A < B
154, 1, 12wbr 3755 . . . 4 wff B < A
1614, 15wo 628 . . 3 wff (A < B B < A)
1713, 16wi 4 . 2 wff ((A · 𝐶) < (B · 𝐶) → (A < B B < A))
188, 17wi 4 1 wff ((A B 𝐶 ℝ) → ((A · 𝐶) < (B · 𝐶) → (A < B B < A)))
Colors of variables: wff set class
This axiom is referenced by:  remulext1  7343
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