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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 4932 | . . . 4 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ (𝐴 − 𝐵)(( ≤ ↾ 𝐸) ∖ I )(𝐶 − 𝐷)) |
3 | brdif 4979 | . . . 4 ⊢ ((𝐴 − 𝐵)(( ≤ ↾ 𝐸) ∖ I )(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))) | |
4 | 2, 3 | bitri 267 | . . 3 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷))) |
5 | ovex 7007 | . . . . 5 ⊢ (𝐶 − 𝐷) ∈ V | |
6 | 5 | brresi 5702 | . . . 4 ⊢ ((𝐴 − 𝐵)( ≤ ↾ 𝐸)(𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ∈ 𝐸 ∧ (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))) |
7 | 6 | anbi1i 615 | . . 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 632 | . . 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 289 | . 2 ⊢ ((𝐴 − 𝐵) < (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∧ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))) |
10 | 5 | ideq 5570 | . . . . 5 ⊢ ((𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐶 − 𝐷)) |
11 | 10 | necon3bbii 3009 | . . . 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 7022 | . . . . . 6 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ( − “ (𝑃 × 𝑃))) |
17 | legso.a | . . . . . 6 ⊢ 𝐸 = ( − “ (𝑃 × 𝑃)) | |
18 | 16, 17 | syl6eleqr 2872 | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ 𝐸) |
19 | 18 | biantrurd 525 | . . . 4 ⊢ (𝜑 → (¬ (𝐴 − 𝐵) I (𝐶 − 𝐷) ↔ ((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)))) |
20 | 11, 19 | syl5rbbr 278 | . . 3 ⊢ (𝜑 → (((𝐴 − 𝐵) ∈ 𝐸 ∧ ¬ (𝐴 − 𝐵) I (𝐶 − 𝐷)) ↔ (𝐴 − 𝐵) ≠ (𝐶 − 𝐷))) |
21 | 20 | anbi2d 620 | . 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 275 | 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 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 ∖ cdif 3821 ⊆ wss 3824 class class class wbr 4926 I cid 5308 × cxp 5402 dom cdm 5404 ↾ cres 5406 “ cima 5407 Fun wfun 6180 ‘cfv 6186 (class class class)co 6975 Basecbs 16338 distcds 16429 TarskiGcstrkg 25934 Itvcitv 25940 ≤Gcleg 26086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pr 5183 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-fv 6194 df-ov 6978 |
This theorem is referenced by: legov3 26102 legso 26103 |
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