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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 487 | . . . . . 6 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → 𝑎 = 〈𝑥, 𝑦〉) | |
3 | 2 | fveq2d 6674 | . . . . 5 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → ( − ‘𝑎) = ( − ‘〈𝑥, 𝑦〉)) |
4 | 1, 3 | eqtr3d 2858 | . . . 4 ⊢ ((((((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑎 = 〈𝑥, 𝑦〉) → 𝐴 = ( − ‘〈𝑥, 𝑦〉)) |
5 | df-ov 7159 | . . . 4 ⊢ (𝑥 − 𝑦) = ( − ‘〈𝑥, 𝑦〉) | |
6 | 4, 5 | syl6eqr 2874 | . . 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 5579 | . . . 4 ⊢ (𝑎 ∈ (𝑃 × 𝑃) ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = 〈𝑥, 𝑦〉) | |
9 | 7, 8 | sylib 220 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑃 × 𝑃)) ∧ ( − ‘𝑎) = 𝐴) → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑎 = 〈𝑥, 𝑦〉) |
10 | 6, 9 | reximddv2 3278 | . 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 2923 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ( − “ (𝑃 × 𝑃))) |
15 | fvelima 6731 | . . 3 ⊢ ((Fun − ∧ 𝐴 ∈ ( − “ (𝑃 × 𝑃))) → ∃𝑎 ∈ (𝑃 × 𝑃)( − ‘𝑎) = 𝐴) | |
16 | 11, 14, 15 | syl2anc 586 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑃 × 𝑃)( − ‘𝑎) = 𝐴) |
17 | 10, 16 | r19.29a 3289 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝐴 = (𝑥 − 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 〈cop 4573 × cxp 5553 “ cima 5558 Fun wfun 6349 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 distcds 16574 TarskiGcstrkg 26216 Itvcitv 26222 ≤Gcleg 26368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 |
This theorem is referenced by: legso 26385 |
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