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Mirrors > Home > MPE Home > Th. List > ltordlem | Structured version Visualization version GIF version |
Description: Lemma for ltord1 10433. (Contributed by Mario Carneiro, 14-Jun-2014.) |
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)) |
Ref | Expression |
---|---|
ltordlem | ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltord.6 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵)) | |
2 | 1 | ralrimivva 2954 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → 𝐴 < 𝐵)) |
3 | breq1 4586 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝑥 < 𝑦 ↔ 𝐶 < 𝑦)) | |
4 | ltord.2 | . . . . 5 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) | |
5 | 4 | breq1d 4593 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 < 𝐵 ↔ 𝑀 < 𝐵)) |
6 | 3, 5 | imbi12d 333 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝑥 < 𝑦 → 𝐴 < 𝐵) ↔ (𝐶 < 𝑦 → 𝑀 < 𝐵))) |
7 | breq2 4587 | . . . 4 ⊢ (𝑦 = 𝐷 → (𝐶 < 𝑦 ↔ 𝐶 < 𝐷)) | |
8 | eqeq1 2614 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐷 ↔ 𝑦 = 𝐷)) | |
9 | ltord.1 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
10 | 9 | eqeq1d 2612 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 = 𝑁 ↔ 𝐵 = 𝑁)) |
11 | 8, 10 | imbi12d 333 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 = 𝐷 → 𝐴 = 𝑁) ↔ (𝑦 = 𝐷 → 𝐵 = 𝑁))) |
12 | ltord.3 | . . . . . 6 ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) | |
13 | 11, 12 | chvarv 2251 | . . . . 5 ⊢ (𝑦 = 𝐷 → 𝐵 = 𝑁) |
14 | 13 | breq2d 4595 | . . . 4 ⊢ (𝑦 = 𝐷 → (𝑀 < 𝐵 ↔ 𝑀 < 𝑁)) |
15 | 7, 14 | imbi12d 333 | . . 3 ⊢ (𝑦 = 𝐷 → ((𝐶 < 𝑦 → 𝑀 < 𝐵) ↔ (𝐶 < 𝐷 → 𝑀 < 𝑁))) |
16 | 6, 15 | rspc2v 3293 | . 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → 𝐴 < 𝐵) → (𝐶 < 𝐷 → 𝑀 < 𝑁))) |
17 | 2, 16 | mpan9 485 | 1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 class class class wbr 4583 ℝcr 9814 < clt 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 |
This theorem is referenced by: ltord1 10433 |
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