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| Mirrors > Home > MPE Home > Th. List > ltgov | Structured version Visualization version GIF version | ||
| Description: Strict "shorter than" geometric relation between segments. (Contributed by Thierry Arnoux, 15-Dec-2019.) | 
| Ref | Expression | 
|---|---|
| legval.p | ⊢ 𝑃 = (Base‘𝐺) | 
| legval.d | ⊢ − = (dist‘𝐺) | 
| legval.i | ⊢ 𝐼 = (Itv‘𝐺) | 
| legval.l | ⊢ ≤ = (≤G‘𝐺) | 
| legval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) | 
| legso.a | ⊢ 𝐸 = ( − “ (𝑃 × 𝑃)) | 
| legso.f | ⊢ (𝜑 → Fun − ) | 
| legso.l | ⊢ < = (( ≤ ↾ 𝐸) ∖ I ) | 
| legso.d | ⊢ (𝜑 → (𝑃 × 𝑃) ⊆ dom − ) | 
| ltgov.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) | 
| ltgov.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) | 
| Ref | Expression | 
|---|---|
| ltgov | ⊢ (𝜑 → ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | legso.l | . . . . 5 ⊢ < = (( ≤ ↾ 𝐸) ∖ I ) | |
| 2 | 1 | breqi 5148 | . . . 4 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ (𝐴 − 𝐵)(( ≤ ↾ 𝐸) ∖ I )(𝐶 − 𝐷)) | 
| 3 | brdif 5195 | . . . 4 ⊢ ((𝐴 − 𝐵)(( ≤ ↾ 𝐸) ∖ I )(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))) | |
| 4 | 2, 3 | bitri 275 | . . 3 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))) | 
| 5 | ovex 7465 | . . . . 5 ⊢ (𝐶 − 𝐷) ∈ V | |
| 6 | 5 | brresi 6005 | . . . 4 ⊢ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ∈ 𝐸 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))) | 
| 7 | 6 | anbi1i 624 | . . 3 ⊢ (((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ (((𝐴 − 𝐵) ∈ 𝐸 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))) | 
| 8 | an21 644 | . . 3 ⊢ ((((𝐴 − 𝐵) ∈ 𝐸 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))) | |
| 9 | 4, 7, 8 | 3bitri 297 | . 2 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))) | 
| 10 | ltgov.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 11 | ltgov.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 12 | legso.f | . . . . . . 7 ⊢ (𝜑 → Fun − ) | |
| 13 | legso.d | . . . . . . 7 ⊢ (𝜑 → (𝑃 × 𝑃) ⊆ dom − ) | |
| 14 | 10, 11, 12, 13 | elovimad 7482 | . . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ( − “ (𝑃 × 𝑃))) | 
| 15 | legso.a | . . . . . 6 ⊢ 𝐸 = ( − “ (𝑃 × 𝑃)) | |
| 16 | 14, 15 | eleqtrrdi 2851 | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ 𝐸) | 
| 17 | 16 | biantrurd 532 | . . . 4 ⊢ (𝜑 → (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))) | 
| 18 | 5 | ideq 5862 | . . . . 5 ⊢ ((𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷)) | 
| 19 | 18 | necon3bbii 2987 | . . . 4 ⊢ (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) | 
| 20 | 17, 19 | bitr3di 286 | . . 3 ⊢ (𝜑 → (((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))) | 
| 21 | 20 | anbi2d 630 | . 2 ⊢ (𝜑 → (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)))) | 
| 22 | 9, 21 | bitrid 283 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∖ cdif 3947 ⊆ wss 3950 class class class wbr 5142 I cid 5576 × cxp 5682 dom cdm 5684 ↾ cres 5686 “ cima 5687 Fun wfun 6554 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 distcds 17307 TarskiGcstrkg 28436 Itvcitv 28442 ≤Gcleg 28591 | 
| 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-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 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-fv 6568 df-ov 7435 | 
| This theorem is referenced by: legov3 28607 legso 28608 | 
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