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| Mirrors > Home > MPE Home > Th. List > leord1 | Structured version Visualization version GIF version | ||
| Description: Infer an ordering relation from a proof in only one direction. (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 |
|---|---|
| leord1 | ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ 𝑀 ≤ 𝑁)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltord.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
| 2 | ltord.3 | . . . . 5 ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) | |
| 3 | ltord.2 | . . . . 5 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) | |
| 4 | ltord.4 | . . . . 5 ⊢ 𝑆 ⊆ ℝ | |
| 5 | ltord.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) | |
| 6 | ltord.6 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵)) | |
| 7 | 1, 2, 3, 4, 5, 6 | ltord1 11789 | . . . 4 ⊢ ((𝜑 ∧ (𝐷 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐷 < 𝐶 ↔ 𝑁 < 𝑀)) |
| 8 | 7 | ancom2s 650 | . . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐷 < 𝐶 ↔ 𝑁 < 𝑀)) |
| 9 | 8 | notbid 318 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (¬ 𝐷 < 𝐶 ↔ ¬ 𝑁 < 𝑀)) |
| 10 | 4 | sseli 3979 | . . . 4 ⊢ (𝐶 ∈ 𝑆 → 𝐶 ∈ ℝ) |
| 11 | 4 | sseli 3979 | . . . 4 ⊢ (𝐷 ∈ 𝑆 → 𝐷 ∈ ℝ) |
| 12 | lenlt 11339 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶)) | |
| 13 | 10, 11, 12 | syl2an 596 | . . 3 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → (𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶)) |
| 14 | 13 | adantl 481 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ ¬ 𝐷 < 𝐶)) |
| 15 | 5 | ralrimiva 3146 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ) |
| 16 | 3 | eleq1d 2826 | . . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝑀 ∈ ℝ)) |
| 17 | 16 | rspccva 3621 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ) |
| 18 | 15, 17 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ) |
| 19 | 18 | adantrr 717 | . . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑀 ∈ ℝ) |
| 20 | 2 | eleq1d 2826 | . . . . . 6 ⊢ (𝑥 = 𝐷 → (𝐴 ∈ ℝ ↔ 𝑁 ∈ ℝ)) |
| 21 | 20 | rspccva 3621 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ) |
| 22 | 15, 21 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ) |
| 23 | 22 | adantrl 716 | . . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑁 ∈ ℝ) |
| 24 | 19, 23 | lenltd 11407 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) |
| 25 | 9, 14, 24 | 3bitr4d 311 | 1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ 𝑀 ≤ 𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 class class class wbr 5143 ℝcr 11154 < clt 11295 ≤ cle 11296 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-pre-lttri 11229 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 |
| This theorem is referenced by: eqord1 11791 leord2 11793 lermxnn0 42962 lermy 42967 |
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