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Mirrors > Home > MPE Home > Th. List > Mathboxes > archiabllem2b | Structured version Visualization version GIF version |
Description: Lemma for archiabl 30827. (Contributed by Thierry Arnoux, 1-May-2018.) |
Ref | Expression |
---|---|
archiabllem.b | ⊢ 𝐵 = (Base‘𝑊) |
archiabllem.0 | ⊢ 0 = (0g‘𝑊) |
archiabllem.e | ⊢ ≤ = (le‘𝑊) |
archiabllem.t | ⊢ < = (lt‘𝑊) |
archiabllem.m | ⊢ · = (.g‘𝑊) |
archiabllem.g | ⊢ (𝜑 → 𝑊 ∈ oGrp) |
archiabllem.a | ⊢ (𝜑 → 𝑊 ∈ Archi) |
archiabllem2.1 | ⊢ + = (+g‘𝑊) |
archiabllem2.2 | ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp) |
archiabllem2.3 | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) |
archiabllem2b.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
archiabllem2b.5 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
archiabllem2b | ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | archiabllem.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
2 | archiabllem.0 | . . 3 ⊢ 0 = (0g‘𝑊) | |
3 | archiabllem.e | . . 3 ⊢ ≤ = (le‘𝑊) | |
4 | archiabllem.t | . . 3 ⊢ < = (lt‘𝑊) | |
5 | archiabllem.m | . . 3 ⊢ · = (.g‘𝑊) | |
6 | archiabllem.g | . . 3 ⊢ (𝜑 → 𝑊 ∈ oGrp) | |
7 | archiabllem.a | . . 3 ⊢ (𝜑 → 𝑊 ∈ Archi) | |
8 | archiabllem2.1 | . . 3 ⊢ + = (+g‘𝑊) | |
9 | archiabllem2.2 | . . 3 ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp) | |
10 | archiabllem2.3 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) | |
11 | archiabllem2b.4 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | archiabllem2b.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | archiabllem2c 30824 | . 2 ⊢ (𝜑 → ¬ (𝑋 + 𝑌) < (𝑌 + 𝑋)) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11 | archiabllem2c 30824 | . 2 ⊢ (𝜑 → ¬ (𝑌 + 𝑋) < (𝑋 + 𝑌)) |
15 | isogrp 30703 | . . . . 5 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd)) | |
16 | 15 | simprbi 499 | . . . 4 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd) |
17 | omndtos 30706 | . . . 4 ⊢ (𝑊 ∈ oMnd → 𝑊 ∈ Toset) | |
18 | 6, 16, 17 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑊 ∈ Toset) |
19 | ogrpgrp 30704 | . . . . 5 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Grp) | |
20 | 6, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Grp) |
21 | 1, 8 | grpcl 18111 | . . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
22 | 20, 11, 12, 21 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
23 | 1, 8 | grpcl 18111 | . . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 + 𝑋) ∈ 𝐵) |
24 | 20, 12, 11, 23 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑌 + 𝑋) ∈ 𝐵) |
25 | 1, 4 | tlt3 30652 | . . 3 ⊢ ((𝑊 ∈ Toset ∧ (𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑌 + 𝑋) ∈ 𝐵) → ((𝑋 + 𝑌) = (𝑌 + 𝑋) ∨ (𝑋 + 𝑌) < (𝑌 + 𝑋) ∨ (𝑌 + 𝑋) < (𝑋 + 𝑌))) |
26 | 18, 22, 24, 25 | syl3anc 1367 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) = (𝑌 + 𝑋) ∨ (𝑋 + 𝑌) < (𝑌 + 𝑋) ∨ (𝑌 + 𝑋) < (𝑋 + 𝑌))) |
27 | 13, 14, 26 | ecase23d 1469 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ w3o 1082 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 lecple 16572 0gc0g 16713 ltcplt 17551 Tosetctos 17643 Grpcgrp 18103 .gcmg 18224 oppgcoppg 18473 oMndcomnd 30698 oGrpcogrp 30699 Archicarchi 30806 |
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-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 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-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 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-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-seq 13371 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-ple 16585 df-0g 16715 df-proset 17538 df-poset 17556 df-plt 17568 df-toset 17644 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mulg 18225 df-oppg 18474 df-omnd 30700 df-ogrp 30701 df-inftm 30807 df-archi 30808 |
This theorem is referenced by: archiabllem2 30826 |
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