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Mirrors > Home > MPE Home > Th. List > xmstrkgc | Structured version Visualization version GIF version |
Description: Any metric space fulfills Tarski's geometry axioms of congruence. (Contributed by Thierry Arnoux, 13-Mar-2019.) |
Ref | Expression |
---|---|
xmstrkgc | ⊢ (𝐺 ∈ ∞MetSp → 𝐺 ∈ TarskiGC) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3514 | . 2 ⊢ (𝐺 ∈ ∞MetSp → 𝐺 ∈ V) | |
2 | eqid 2823 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | eqid 2823 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | 2, 3 | xmssym 23077 | . . . . 5 ⊢ ((𝐺 ∈ ∞MetSp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝑥(dist‘𝐺)𝑦) = (𝑦(dist‘𝐺)𝑥)) |
5 | 4 | 3expb 1116 | . . . 4 ⊢ ((𝐺 ∈ ∞MetSp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(dist‘𝐺)𝑦) = (𝑦(dist‘𝐺)𝑥)) |
6 | 5 | ralrimivva 3193 | . . 3 ⊢ (𝐺 ∈ ∞MetSp → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(dist‘𝐺)𝑦) = (𝑦(dist‘𝐺)𝑥)) |
7 | simpl 485 | . . . . . . . 8 ⊢ ((𝐺 ∈ ∞MetSp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝐺 ∈ ∞MetSp) | |
8 | simpr3 1192 | . . . . . . . 8 ⊢ ((𝐺 ∈ ∞MetSp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → 𝑧 ∈ (Base‘𝐺)) | |
9 | equid 2019 | . . . . . . . . 9 ⊢ 𝑧 = 𝑧 | |
10 | 2, 3 | xmseq0 23076 | . . . . . . . . 9 ⊢ ((𝐺 ∈ ∞MetSp ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → ((𝑧(dist‘𝐺)𝑧) = 0 ↔ 𝑧 = 𝑧)) |
11 | 9, 10 | mpbiri 260 | . . . . . . . 8 ⊢ ((𝐺 ∈ ∞MetSp ∧ 𝑧 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑧(dist‘𝐺)𝑧) = 0) |
12 | 7, 8, 8, 11 | syl3anc 1367 | . . . . . . 7 ⊢ ((𝐺 ∈ ∞MetSp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → (𝑧(dist‘𝐺)𝑧) = 0) |
13 | 12 | eqeq2d 2834 | . . . . . 6 ⊢ ((𝐺 ∈ ∞MetSp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥(dist‘𝐺)𝑦) = (𝑧(dist‘𝐺)𝑧) ↔ (𝑥(dist‘𝐺)𝑦) = 0)) |
14 | 2, 3 | xmseq0 23076 | . . . . . . 7 ⊢ ((𝐺 ∈ ∞MetSp ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ((𝑥(dist‘𝐺)𝑦) = 0 ↔ 𝑥 = 𝑦)) |
15 | 14 | 3adant3r3 1180 | . . . . . 6 ⊢ ((𝐺 ∈ ∞MetSp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥(dist‘𝐺)𝑦) = 0 ↔ 𝑥 = 𝑦)) |
16 | 13, 15 | bitrd 281 | . . . . 5 ⊢ ((𝐺 ∈ ∞MetSp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥(dist‘𝐺)𝑦) = (𝑧(dist‘𝐺)𝑧) ↔ 𝑥 = 𝑦)) |
17 | 16 | biimpd 231 | . . . 4 ⊢ ((𝐺 ∈ ∞MetSp ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺))) → ((𝑥(dist‘𝐺)𝑦) = (𝑧(dist‘𝐺)𝑧) → 𝑥 = 𝑦)) |
18 | 17 | ralrimivvva 3194 | . . 3 ⊢ (𝐺 ∈ ∞MetSp → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(dist‘𝐺)𝑦) = (𝑧(dist‘𝐺)𝑧) → 𝑥 = 𝑦)) |
19 | 6, 18 | jca 514 | . 2 ⊢ (𝐺 ∈ ∞MetSp → (∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(dist‘𝐺)𝑦) = (𝑦(dist‘𝐺)𝑥) ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(dist‘𝐺)𝑦) = (𝑧(dist‘𝐺)𝑧) → 𝑥 = 𝑦))) |
20 | eqid 2823 | . . 3 ⊢ (Itv‘𝐺) = (Itv‘𝐺) | |
21 | 2, 3, 20 | istrkgc 26242 | . 2 ⊢ (𝐺 ∈ TarskiGC ↔ (𝐺 ∈ V ∧ (∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(dist‘𝐺)𝑦) = (𝑦(dist‘𝐺)𝑥) ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑥(dist‘𝐺)𝑦) = (𝑧(dist‘𝐺)𝑧) → 𝑥 = 𝑦)))) |
22 | 1, 19, 21 | sylanbrc 585 | 1 ⊢ (𝐺 ∈ ∞MetSp → 𝐺 ∈ TarskiGC) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ‘cfv 6357 (class class class)co 7158 0cc0 10539 Basecbs 16485 distcds 16576 ∞MetSpcxms 22929 TarskiGCcstrkgc 26219 Itvcitv 26224 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-xms 22932 df-trkgc 26236 |
This theorem is referenced by: (None) |
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