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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 4659 | . . . 4 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ (𝐴 − 𝐵)(( ≤ ↾ 𝐸) ∖ I )(𝐶 − 𝐷)) |
3 | brdif 4705 | . . . 4 ⊢ ((𝐴 − 𝐵)(( ≤ ↾ 𝐸) ∖ I )(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))) | |
4 | 2, 3 | bitri 264 | . . 3 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))) |
5 | ovex 6678 | . . . . 5 ⊢ (𝐶 − 𝐷) ∈ V | |
6 | 5 | brres 5402 | . . . 4 ⊢ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ∈ 𝐸)) |
7 | 6 | anbi1i 731 | . . 3 ⊢ (((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ∈ 𝐸) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))) |
8 | anass 681 | . . 3 ⊢ ((((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ (𝐴 − 𝐵) ∈ 𝐸) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))) | |
9 | 4, 7, 8 | 3bitri 286 | . 2 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))) |
10 | 5 | ideq 5274 | . . . . 5 ⊢ ((𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷)) |
11 | 10 | necon3bbii 2841 | . . . 4 ⊢ (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷)) |
12 | ltgov.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
13 | ltgov.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
14 | legso.f | . . . . . . 7 ⊢ (𝜑 → Fun − ) | |
15 | legso.d | . . . . . . 7 ⊢ (𝜑 → (𝑃 × 𝑃) ⊆ dom − ) | |
16 | 12, 13, 14, 15 | elovimad 6693 | . . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ( − “ (𝑃 × 𝑃))) |
17 | legso.a | . . . . . 6 ⊢ 𝐸 = ( − “ (𝑃 × 𝑃)) | |
18 | 16, 17 | syl6eleqr 2712 | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ 𝐸) |
19 | 18 | biantrurd 529 | . . . 4 ⊢ (𝜑 → (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))) |
20 | 11, 19 | syl5rbbr 275 | . . 3 ⊢ (𝜑 → (((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))) |
21 | 20 | anbi2d 740 | . 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 | syl5bb 272 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ⊆ wss 3574 class class class wbr 4653 I cid 5023 × cxp 5112 dom cdm 5114 ↾ cres 5116 “ cima 5117 Fun wfun 5882 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 ≤Gcleg 25477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-ov 6653 |
This theorem is referenced by: legov3 25493 legso 25494 |
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