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| Mirrors > Home > MPE Home > Th. List > ltord2 | 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) | 
| ltord2.6 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴)) | 
| Ref | Expression | 
|---|---|
| ltord2 | ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 ↔ 𝑁 < 𝑀)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ltord.1 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
| 2 | 1 | negeqd 11503 | . . 3 ⊢ (𝑥 = 𝑦 → -𝐴 = -𝐵) | 
| 3 | ltord.2 | . . . 4 ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀) | |
| 4 | 3 | negeqd 11503 | . . 3 ⊢ (𝑥 = 𝐶 → -𝐴 = -𝑀) | 
| 5 | ltord.3 | . . . 4 ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁) | |
| 6 | 5 | negeqd 11503 | . . 3 ⊢ (𝑥 = 𝐷 → -𝐴 = -𝑁) | 
| 7 | ltord.4 | . . 3 ⊢ 𝑆 ⊆ ℝ | |
| 8 | ltord.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ) | |
| 9 | 8 | renegcld 11691 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → -𝐴 ∈ ℝ) | 
| 10 | ltord2.6 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴)) | |
| 11 | 8 | ralrimiva 3145 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ) | 
| 12 | 1 | eleq1d 2825 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℝ ↔ 𝐵 ∈ ℝ)) | 
| 13 | 12 | rspccva 3620 | . . . . . . 7 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝑦 ∈ 𝑆) → 𝐵 ∈ ℝ) | 
| 14 | 11, 13 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐵 ∈ ℝ) | 
| 15 | 14 | adantrl 716 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐵 ∈ ℝ) | 
| 16 | 8 | adantrr 717 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝐴 ∈ ℝ) | 
| 17 | ltneg 11764 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 ↔ -𝐴 < -𝐵)) | |
| 18 | 15, 16, 17 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐵 < 𝐴 ↔ -𝐴 < -𝐵)) | 
| 19 | 10, 18 | sylibd 239 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → -𝐴 < -𝐵)) | 
| 20 | 2, 4, 6, 7, 9, 19 | ltord1 11790 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 ↔ -𝑀 < -𝑁)) | 
| 21 | 5 | eleq1d 2825 | . . . . . 6 ⊢ (𝑥 = 𝐷 → (𝐴 ∈ ℝ ↔ 𝑁 ∈ ℝ)) | 
| 22 | 21 | rspccva 3620 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ) | 
| 23 | 11, 22 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑆) → 𝑁 ∈ ℝ) | 
| 24 | 23 | adantrl 716 | . . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑁 ∈ ℝ) | 
| 25 | 3 | eleq1d 2825 | . . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ ℝ ↔ 𝑀 ∈ ℝ)) | 
| 26 | 25 | rspccva 3620 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝑆 𝐴 ∈ ℝ ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ) | 
| 27 | 11, 26 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑆) → 𝑀 ∈ ℝ) | 
| 28 | 27 | adantrr 717 | . . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → 𝑀 ∈ ℝ) | 
| 29 | ltneg 11764 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑁 < 𝑀 ↔ -𝑀 < -𝑁)) | |
| 30 | 24, 28, 29 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝑁 < 𝑀 ↔ -𝑀 < -𝑁)) | 
| 31 | 20, 30 | bitr4d 282 | 1 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 ↔ 𝑁 < 𝑀)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ⊆ wss 3950 class class class wbr 5142 ℝcr 11155 < clt 11296 -cneg 11494 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 | 
| This theorem is referenced by: (None) | 
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