Theorem ltordlem 8554

Description: Lemma for eqord1 8555. (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 2587	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → 𝐴 < 𝐵))
3		breq1 4046	. . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 < 𝑦 ↔ 𝐶 < 𝑦))
4		ltord.2	. . . . 5 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)
5	4	breq1d 4053	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 < 𝐵 ↔ 𝑀 < 𝐵))
6	3, 5	imbi12d 234	. . 3 ⊢ (𝑥 = 𝐶 → ((𝑥 < 𝑦 → 𝐴 < 𝐵) ↔ (𝐶 < 𝑦 → 𝑀 < 𝐵)))
7		breq2 4047	. . . 4 ⊢ (𝑦 = 𝐷 → (𝐶 < 𝑦 ↔ 𝐶 < 𝐷))
8		eqeq1 2211	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐷 ↔ 𝑦 = 𝐷))
9		ltord.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
10	9	eqeq1d 2213	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝑁 ↔ 𝐵 = 𝑁))
11	8, 10	imbi12d 234	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐷 → 𝐴 = 𝑁) ↔ (𝑦 = 𝐷 → 𝐵 = 𝑁)))
12		ltord.3	. . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)
13	11, 12	chvarv 1964	. . . . 5 ⊢ (𝑦 = 𝐷 → 𝐵 = 𝑁)
14	13	breq2d 4055	. . . 4 ⊢ (𝑦 = 𝐷 → (𝑀 < 𝐵 ↔ 𝑀 < 𝑁))
15	7, 14	imbi12d 234	. . 3 ⊢ (𝑦 = 𝐷 → ((𝐶 < 𝑦 → 𝑀 < 𝐵) ↔ (𝐶 < 𝐷 → 𝑀 < 𝑁)))
16	6, 15	rspc2v 2889	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → 𝐴 < 𝐵) → (𝐶 < 𝐷 → 𝑀 < 𝑁)))
17	2, 16	mpan9 281	1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372   ∈ wcel 2175  ∀wral 2483   ⊆ wss 3165   class class class wbr 4043  ℝcr 7923   < clt 8106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044

This theorem is referenced by:  eqord1  8555