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 ) )