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Mirrors > Home > MPE Home > Th. List > legid | Structured version Visualization version GIF version |
Description: Reflexivity of the less-than relationship. Proposition 5.7 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) |
Ref | Expression |
---|---|
legid | ⊢ (𝜑 → (𝐴 − 𝐵) ≤ (𝐴 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | legid.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 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 | legid.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 2, 3, 4, 5, 6, 1 | tgbtwntriv2 26433 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
8 | eqidd 2739 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐴 − 𝐵)) | |
9 | eleq1 2820 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴𝐼𝐵) ↔ 𝐵 ∈ (𝐴𝐼𝐵))) | |
10 | oveq2 7178 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 − 𝑥) = (𝐴 − 𝐵)) | |
11 | 10 | eqeq2d 2749 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 − 𝐵) = (𝐴 − 𝑥) ↔ (𝐴 − 𝐵) = (𝐴 − 𝐵))) |
12 | 9, 11 | anbi12d 634 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ (𝐴 − 𝐵) = (𝐴 − 𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ (𝐴 − 𝐵) = (𝐴 − 𝐵)))) |
13 | 12 | rspcev 3526 | . . 3 ⊢ ((𝐵 ∈ 𝑃 ∧ (𝐵 ∈ (𝐴𝐼𝐵) ∧ (𝐴 − 𝐵) = (𝐴 − 𝐵))) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ (𝐴 − 𝐵) = (𝐴 − 𝑥))) |
14 | 1, 7, 8, 13 | syl12anc 836 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ (𝐴 − 𝐵) = (𝐴 − 𝑥))) |
15 | legval.l | . . 3 ⊢ ≤ = (≤G‘𝐺) | |
16 | 2, 3, 4, 15, 5, 6, 1, 6, 1 | legov 26531 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐴 − 𝐵) ↔ ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ (𝐴 − 𝐵) = (𝐴 − 𝑥)))) |
17 | 14, 16 | mpbird 260 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ≤ (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∃wrex 3054 class class class wbr 5030 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 distcds 16677 TarskiGcstrkg 26376 Itvcitv 26382 ≤Gcleg 26528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-oadd 8135 df-er 8320 df-pm 8440 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-dju 9403 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-xnn0 12049 df-z 12063 df-uz 12325 df-fz 12982 df-fzo 13125 df-hash 13783 df-word 13956 df-concat 14012 df-s1 14039 df-s2 14299 df-s3 14300 df-trkgc 26394 df-trkgb 26395 df-trkgcb 26396 df-trkg 26399 df-cgrg 26457 df-leg 26529 |
This theorem is referenced by: legtrid 26537 legov3 26544 |
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