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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 784 | . . . . 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 6911 | . . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → ( − ‘𝑎) = ( − ‘〈𝑥, 𝑦〉)) |
4 | 1, 3 | eqtr3d 2777 | . . . 4 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → 𝐴 = ( − ‘〈𝑥, 𝑦〉)) |
5 | df-ov 7434 | . . . 4 ⊢ (𝑥 − 𝑦) = ( − ‘〈𝑥, 𝑦〉) | |
6 | 4, 5 | eqtr4di 2793 | . . 3 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → 𝐴 = (𝑥 − 𝑦)) |
7 | simplr 769 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃)) | |
8 | elxp2 5713 | . . . 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 3213 | . 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 2849 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ( − “ (𝑃 × 𝑃))) |
15 | fvelima 6974 | . . 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 3160 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝐴 = (𝑥 − 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 〈cop 4637 × cxp 5687 “ cima 5692 Fun wfun 6557 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 distcds 17307 TarskiGcstrkg 28450 Itvcitv 28456 ≤Gcleg 28605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 |
This theorem is referenced by: legso 28622 |
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