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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 782 | . . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → ( − ‘𝑎) = 𝐴) | |
2 | simpr 485 | . . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → 𝑎 = 〈𝑥, 𝑦〉) | |
3 | 2 | fveq2d 6882 | . . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → ( − ‘𝑎) = ( − ‘〈𝑥, 𝑦〉)) |
4 | 1, 3 | eqtr3d 2773 | . . . 4 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → 𝐴 = ( − ‘〈𝑥, 𝑦〉)) |
5 | df-ov 7396 | . . . 4 ⊢ (𝑥 − 𝑦) = ( − ‘〈𝑥, 𝑦〉) | |
6 | 4, 5 | eqtr4di 2789 | . . 3 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → 𝐴 = (𝑥 − 𝑦)) |
7 | simplr 767 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → 𝑎 ∈ (𝑃 × 𝑃)) | |
8 | elxp2 5693 | . . . 4 ⊢ (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = 〈𝑥, 𝑦〉) | |
9 | 7, 8 | sylib 217 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = 〈𝑥, 𝑦〉) |
10 | 6, 9 | reximddv2 3211 | . 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 2842 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ( − “ (𝑃 × 𝑃))) |
15 | fvelima 6944 | . . 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 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 〈cop 4628 × cxp 5667 “ cima 5672 Fun wfun 6526 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 distcds 17188 TarskiGcstrkg 27543 Itvcitv 27549 ≤Gcleg 27698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fv 6540 df-ov 7396 |
This theorem is referenced by: legso 27715 |
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