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Theorem angmgm 29277

Description: The angle addition magma is a magma. (Contributed by Thierry Arnoux, 31-Aug-2026.)

**Hypotheses**
Ref	Expression
angmgmval.p	⊢ 𝑃 = (Base‘𝐺)
angmgmval.a	⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
angmgmval.i	⊢ 𝐼 = (Itv‘𝐺)
angmgmval.d	⊢ − = (dist‘𝐺)
angmgmval.c	⊢ ∼ = (cgrA‘𝐺)
angmgmval.l	⊢ 𝐿 = (LineG‘𝐺)
angmgmval.o	⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))
angmgmval.j	⊢ 𝐽 = (AngMgm‘𝐺)
angmgm.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
angmgm.1	⊢ (𝜑 → 2 ≤ (♯‘𝑃))

**Assertion**
Ref	Expression
angmgm	⊢ (𝜑 → 𝐽 ∈ Mgm)

Distinct variable groups: 𝐴,𝑒,𝑓,𝑠 𝑒,𝐺,𝑓,𝑠 𝑃,𝑑,𝑒,𝑓,𝑠 + ,𝑑,𝑒,𝑓,𝑠 − ,𝑑,𝑒,𝑓,𝑠 ∼ ,𝑑,𝑒,𝑓,𝑠 𝐴,𝑑 𝐺,𝑑 𝐼,𝑑,𝑒,𝑓,𝑠 𝑒,𝐽,𝑓 𝐿,𝑑,𝑒,𝑓,𝑠 𝜑,𝑑,𝑒,𝑓,𝑠 Allowed substitution hints: 𝐽(𝑠, 𝑑)

Proof of Theorem angmgm Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		angmgmval.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		angmgmval.a	. . . 4 ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
3		angmgmval.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
4		angmgmval.d	. . . 4 ⊢ − = (dist‘𝐺)
5		angmgmval.c	. . . 4 ⊢ ∼ = (cgrA‘𝐺)
6		angmgmval.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
7		angmgmval.o	. . . 4 ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))
8		angmgmval.j	. . . 4 ⊢ 𝐽 = (AngMgm‘𝐺)
9		angmgm.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
10	9	ad3antrrr 743	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝐺 ∈ TarskiG)
11		simpllr 788	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑥 ∈ 𝑃)
12		simplr 781	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ 𝑃)
13		simpr 490	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑥 ≠ 𝑦)
14	13	necomd 3012	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑦 ≠ 𝑥)
15	12, 14	eldifsnd 4753	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ (𝑃 ∖ {𝑥}))
16	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 15	angmgmlem 29275	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → (𝐽 ∈ Mgm ∧ [⟨“𝑥𝑦𝑥”⟩] ∼ = (0_g‘𝐽)))
17	16	simpld 500	. 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝐽 ∈ Mgm)
18		angmgm.1	. . 3 ⊢ (𝜑 → 2 ≤ (♯‘𝑃))
19	1, 4, 3, 9, 18	tglowdim1 28843	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑥 ≠ 𝑦)
20	17, 19	r19.29vva 3224	1 ⊢ (𝜑 → 𝐽 ∈ Mgm)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 {crab 3414 ∩ cin 3901 ∅c0 4282 ifcif 4485 class class class wbr 5107 ‘cfv 6537 ℩crio 7372 (class class class)co 7416 ∈ cmpo 7418 [cec 8697 ↑_m cmap 8829 0cc0 11127 1c1 11128 ≤ cle 11271 2c2 12322 3c3 12323 ..^cfzo 13711 ♯chash 14396 ⟨“cs3 14915 Basecbs 17305 distcds 17355 0_gc0g 17528 Mgmcmgm 18732 TarskiGcstrkg 28769 Itvcitv 28775 LineGclng 28776 cgrAccgra 29194 AngMgmcangmgm 29253

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-oadd 8462 df-er 8699 df-ec 8701 df-qs 8705 df-map 8831 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9415 df-inf 9416 df-dju 9909 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-xnn0 12605 df-z 12619 df-dec 12740 df-uz 12891 df-fz 13564 df-fzo 13712 df-hash 14397 df-word 14581 df-concat 14638 df-s1 14665 df-s2 14921 df-s3 14922 df-struct 17243 df-slot 17278 df-ndx 17290 df-base 17306 df-plusg 17359 df-mulr 17360 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ds 17368 df-0g 17530 df-imas 17598 df-qus 17599 df-mgm 18734 df-trkgc 28790 df-trkgb 28791 df-trkgcb 28792 df-trkgld 28794 df-trkg 28795 df-cgrg 28854 df-ismt 28876 df-leg 28926 df-hlg 28944 df-mir 29005 df-rag 29049 df-perpg 29051 df-hpg 29116 df-mid 29159 df-lmi 29160 df-cgra 29195 df-angmgm 29254

This theorem is used by: (None)

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