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Theorem ltaprlem 7327
 Description: Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.)
Assertion
Ref Expression
ltaprlem (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Proof of Theorem ltaprlem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ltexpri 7322 . . . 4 (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
21adantr 272 . . 3 ((𝐴<P 𝐵𝐶P) → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
3 simplr 500 . . . . . 6 (((𝐴<P 𝐵𝐶P) ∧ (𝑥P ∧ (𝐴 +P 𝑥) = 𝐵)) → 𝐶P)
4 ltrelpr 7214 . . . . . . . . . 10 <P ⊆ (P × P)
54brel 4529 . . . . . . . . 9 (𝐴<P 𝐵 → (𝐴P𝐵P))
65simpld 111 . . . . . . . 8 (𝐴<P 𝐵𝐴P)
76adantr 272 . . . . . . 7 ((𝐴<P 𝐵𝐶P) → 𝐴P)
87adantr 272 . . . . . 6 (((𝐴<P 𝐵𝐶P) ∧ (𝑥P ∧ (𝐴 +P 𝑥) = 𝐵)) → 𝐴P)
9 addclpr 7246 . . . . . 6 ((𝐶P𝐴P) → (𝐶 +P 𝐴) ∈ P)
103, 8, 9syl2anc 406 . . . . 5 (((𝐴<P 𝐵𝐶P) ∧ (𝑥P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P 𝐴) ∈ P)
11 simprl 501 . . . . 5 (((𝐴<P 𝐵𝐶P) ∧ (𝑥P ∧ (𝐴 +P 𝑥) = 𝐵)) → 𝑥P)
12 ltaddpr 7306 . . . . 5 (((𝐶 +P 𝐴) ∈ P𝑥P) → (𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥))
1310, 11, 12syl2anc 406 . . . 4 (((𝐴<P 𝐵𝐶P) ∧ (𝑥P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥))
14 addassprg 7288 . . . . . 6 ((𝐶P𝐴P𝑥P) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)))
153, 8, 11, 14syl3anc 1184 . . . . 5 (((𝐴<P 𝐵𝐶P) ∧ (𝑥P ∧ (𝐴 +P 𝑥) = 𝐵)) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)))
16 oveq2 5714 . . . . . 6 ((𝐴 +P 𝑥) = 𝐵 → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵))
1716ad2antll 478 . . . . 5 (((𝐴<P 𝐵𝐶P) ∧ (𝑥P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵))
1815, 17eqtrd 2132 . . . 4 (((𝐴<P 𝐵𝐶P) ∧ (𝑥P ∧ (𝐴 +P 𝑥) = 𝐵)) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))
1913, 18breqtrd 3899 . . 3 (((𝐴<P 𝐵𝐶P) ∧ (𝑥P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))
202, 19rexlimddv 2513 . 2 ((𝐴<P 𝐵𝐶P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))
2120expcom 115 1 (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1299   ∈ wcel 1448  ∃wrex 2376   class class class wbr 3875  (class class class)co 5706  Pcnp 7000   +P cpp 7002
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