Theorem ltordlem 11764

Description: Lemma for ltord1 11765. (Contributed by Mario Carneiro, 14-Jun-2014.)

Hypotheses

Ref	Expression
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

Ref	Expression
ltordlem	⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑦,𝐶   𝑥,𝐷,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦)   𝐵(𝑦)

Proof of Theorem ltordlem

Step	Hyp	Ref	Expression
1		ltord.6	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵))
2	1	ralrimivva 3205	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → 𝐴 < 𝐵))
3		breq1 5106	. . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 < 𝑦 ↔ 𝐶 < 𝑦))
4		ltord.2	. . . . 5 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)
5	4	breq1d 5113	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 < 𝐵 ↔ 𝑀 < 𝐵))
6	3, 5	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥 < 𝑦 → 𝐴 < 𝐵) ↔ (𝐶 < 𝑦 → 𝑀 < 𝐵)))
7		breq2 5107	. . . 4 ⊢ (𝑦 = 𝐷 → (𝐶 < 𝑦 ↔ 𝐶 < 𝐷))
8		eqeq1 2764	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐷 ↔ 𝑦 = 𝐷))
9		ltord.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
10	9	eqeq1d 2762	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝑁 ↔ 𝐵 = 𝑁))
11	8, 10	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐷 → 𝐴 = 𝑁) ↔ (𝑦 = 𝐷 → 𝐵 = 𝑁)))
12		ltord.3	. . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)
13	11, 12	chvarvv 2022	. . . . 5 ⊢ (𝑦 = 𝐷 → 𝐵 = 𝑁)
14	13	breq2d 5115	. . . 4 ⊢ (𝑦 = 𝐷 → (𝑀 < 𝐵 ↔ 𝑀 < 𝑁))
15	7, 14	imbi12d 347	. . 3 ⊢ (𝑦 = 𝐷 → ((𝐶 < 𝑦 → 𝑀 < 𝐵) ↔ (𝐶 < 𝐷 → 𝑀 < 𝑁)))
16	6, 15	rspc2v 3587	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → 𝐴 < 𝐵) → (𝐶 < 𝐷 → 𝑀 < 𝑁)))
17	2, 16	mpan9 516	1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899   class class class wbr 5103  ℝcr 11124   < clt 11268

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104

This theorem is used by:  ltord1  11765