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Metamath Proof Explorer

Theorem ltordlem

Description: Lemma for ltord1 . (Contributed by Mario Carneiro, 14-Jun-2014)

Ref Expression
Hypotheses ltord.1 ⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
ltord.2 ⊢ ( 𝑥 = 𝐶 → 𝐴 = 𝑀 )
ltord.3 ⊢ ( 𝑥 = 𝐷 → 𝐴 = 𝑁 )
ltord.4 ⊢ 𝑆 ⊆ ℝ
ltord.5 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → 𝐴 ∈ ℝ )
ltord.6 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 < 𝑦 → 𝐴 < 𝐵 ) )
Assertion ltordlem ( ( 𝜑 ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝐶 < 𝐷 → 𝑀 < 𝑁 ) )

Proof

Step Hyp Ref Expression
1 ltord.1 ⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
2 ltord.2 ⊢ ( 𝑥 = 𝐶 → 𝐴 = 𝑀 )
3 ltord.3 ⊢ ( 𝑥 = 𝐷 → 𝐴 = 𝑁 )
4 ltord.4 ⊢ 𝑆 ⊆ ℝ
5 ltord.5 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → 𝐴 ∈ ℝ )
6 ltord.6 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 < 𝑦 → 𝐴 < 𝐵 ) )
7 6 ralrimivva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 < 𝑦 → 𝐴 < 𝐵 ) )
8 breq1 ⊢ ( 𝑥 = 𝐶 → ( 𝑥 < 𝑦 ↔ 𝐶 < 𝑦 ) )
9 2 breq1d ⊢ ( 𝑥 = 𝐶 → ( 𝐴 < 𝐵 ↔ 𝑀 < 𝐵 ) )
10 8 9 imbi12d ⊢ ( 𝑥 = 𝐶 → ( ( 𝑥 < 𝑦 → 𝐴 < 𝐵 ) ↔ ( 𝐶 < 𝑦 → 𝑀 < 𝐵 ) ) )
11 breq2 ⊢ ( 𝑦 = 𝐷 → ( 𝐶 < 𝑦 ↔ 𝐶 < 𝐷 ) )
12 eqeq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝐷 ↔ 𝑦 = 𝐷 ) )
13 1 eqeq1d ⊢ ( 𝑥 = 𝑦 → ( 𝐴 = 𝑁 ↔ 𝐵 = 𝑁 ) )
14 12 13 imbi12d ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 = 𝐷 → 𝐴 = 𝑁 ) ↔ ( 𝑦 = 𝐷 → 𝐵 = 𝑁 ) ) )
15 14 3 chvarvv ⊢ ( 𝑦 = 𝐷 → 𝐵 = 𝑁 )
16 15 breq2d ⊢ ( 𝑦 = 𝐷 → ( 𝑀 < 𝐵 ↔ 𝑀 < 𝑁 ) )
17 11 16 imbi12d ⊢ ( 𝑦 = 𝐷 → ( ( 𝐶 < 𝑦 → 𝑀 < 𝐵 ) ↔ ( 𝐶 < 𝐷 → 𝑀 < 𝑁 ) ) )
18 10 17 rspc2v ⊢ ( ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 < 𝑦 → 𝐴 < 𝐵 ) → ( 𝐶 < 𝐷 → 𝑀 < 𝑁 ) ) )
19 7 18 mpan9 ⊢ ( ( 𝜑 ∧ ( 𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆 ) ) → ( 𝐶 < 𝐷 → 𝑀 < 𝑁 ) )