|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > ltgseg | Structured version Visualization version GIF version | ||
| Description: The set 𝐸 denotes the possible values of the congruence. (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 − ) | 
| ltgseg.p | ⊢ (𝜑 → 𝐴 ∈ 𝐸) | 
| Ref | Expression | 
|---|---|
| ltgseg | ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝐴 = (𝑥 − 𝑦)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simp-4r 783 | . . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → ( − ‘𝑎) = 𝐴) | |
| 2 | simpr 484 | . . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → 𝑎 = 〈𝑥, 𝑦〉) | |
| 3 | 2 | fveq2d 6909 | . . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → ( − ‘𝑎) = ( − ‘〈𝑥, 𝑦〉)) | 
| 4 | 1, 3 | eqtr3d 2778 | . . . 4 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → 𝐴 = ( − ‘〈𝑥, 𝑦〉)) | 
| 5 | df-ov 7435 | . . . 4 ⊢ (𝑥 − 𝑦) = ( − ‘〈𝑥, 𝑦〉) | |
| 6 | 4, 5 | eqtr4di 2794 | . . 3 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → 𝐴 = (𝑥 − 𝑦)) | 
| 7 | simplr 768 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃)) | |
| 8 | elxp2 5708 | . . . 4 ⊢ (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = 〈𝑥, 𝑦〉) | |
| 9 | 7, 8 | sylib 218 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = 〈𝑥, 𝑦〉) | 
| 10 | 6, 9 | reximddv2 3214 | . 2 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝐴 = (𝑥 − 𝑦)) | 
| 11 | legso.f | . . 3 ⊢ (𝜑 → Fun − ) | |
| 12 | ltgseg.p | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐸) | |
| 13 | legso.a | . . . 4 ⊢ 𝐸 = ( − “ (𝑃 × 𝑃)) | |
| 14 | 12, 13 | eleqtrdi 2850 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ( − “ (𝑃 × 𝑃))) | 
| 15 | fvelima 6973 | . . 3 ⊢ ((Fun − ∧ 𝐴 ∈ ( − “ (𝑃 × 𝑃))) → ∃𝑎 ∈ (𝑃 × 𝑃)( − ‘𝑎) = 𝐴) | |
| 16 | 11, 14, 15 | syl2anc 584 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑃 × 𝑃)( − ‘𝑎) = 𝐴) | 
| 17 | 10, 16 | r19.29a 3161 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝐴 = (𝑥 − 𝑦)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 〈cop 4631 × cxp 5682 “ 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-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-fv 6568 df-ov 7435 | 
| This theorem is referenced by: legso 28608 | 
| Copyright terms: Public domain | W3C validator |