![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > leg0 | Structured version Visualization version GIF version |
Description: Degenerated (zero-length) segments are minimal. Proposition 5.11 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-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) |
legid.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
legid.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
legtrd.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
legtrd.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
Ref | Expression |
---|---|
leg0 | ⊢ (𝜑 → (𝐴 − 𝐴) ≤ (𝐶 − 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | legtrd.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
2 | legval.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
3 | legval.d | . . . 4 ⊢ − = (dist‘𝐺) | |
4 | legval.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | legval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | legtrd.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
7 | 2, 3, 4, 5, 1, 6 | tgbtwntriv1 25963 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐶𝐼𝐷)) |
8 | legid.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | 2, 3, 4, 5, 8, 1 | tgcgrtriv 25956 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐴) = (𝐶 − 𝐶)) |
10 | eleq1 2872 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ (𝐶𝐼𝐷) ↔ 𝐶 ∈ (𝐶𝐼𝐷))) | |
11 | oveq2 7031 | . . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐶 − 𝑥) = (𝐶 − 𝐶)) | |
12 | 11 | eqeq2d 2807 | . . . . 5 ⊢ (𝑥 = 𝐶 → ((𝐴 − 𝐴) = (𝐶 − 𝑥) ↔ (𝐴 − 𝐴) = (𝐶 − 𝐶))) |
13 | 10, 12 | anbi12d 630 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐴) = (𝐶 − 𝑥)) ↔ (𝐶 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐴) = (𝐶 − 𝐶)))) |
14 | 13 | rspcev 3561 | . . 3 ⊢ ((𝐶 ∈ 𝑃 ∧ (𝐶 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐴) = (𝐶 − 𝐶))) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐴) = (𝐶 − 𝑥))) |
15 | 1, 7, 9, 14 | syl12anc 833 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐴) = (𝐶 − 𝑥))) |
16 | legval.l | . . 3 ⊢ ≤ = (≤G‘𝐺) | |
17 | 2, 3, 4, 16, 5, 8, 8, 1, 6 | legov 26057 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐴) ≤ (𝐶 − 𝐷) ↔ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐶𝐼𝐷) ∧ (𝐴 − 𝐴) = (𝐶 − 𝑥)))) |
18 | 15, 17 | mpbird 258 | 1 ⊢ (𝜑 → (𝐴 − 𝐴) ≤ (𝐶 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∃wrex 3108 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 Basecbs 16316 distcds 16407 TarskiGcstrkg 25902 Itvcitv 25908 ≤Gcleg 26054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-pm 8266 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-dju 9183 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-n0 11752 df-xnn0 11822 df-z 11836 df-uz 12098 df-fz 12747 df-fzo 12888 df-hash 13545 df-word 13712 df-concat 13773 df-s1 13798 df-s2 14050 df-s3 14051 df-trkgc 25920 df-trkgb 25921 df-trkgcb 25922 df-trkg 25925 df-cgrg 25983 df-leg 26055 |
This theorem is referenced by: legeq 26065 |
Copyright terms: Public domain | W3C validator |