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Theorem angmgm 29331

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 3010	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑦 ≠ 𝑥)
15	12, 14	eldifsnd 4749	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ 𝑥 ≠ 𝑦) → 𝑦 ∈ (𝑃 ∖ {𝑥}))
16	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 15	angmgmlem 29329	. . 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 28897	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 𝑥 ≠ 𝑦)
20	17, 19	r19.29vva 3222	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 2955 {crab 3412 ∩ cin 3897 ∅c0 4278 ifcif 4481 class class class wbr 5102 ‘cfv 6527 ℩crio 7364 (class class class)co 7408 ∈ cmpo 7410 [cec 8693 ↑_m cmap 8825 0cc0 11172 1c1 11173 ≤ cle 11316 2c2 12367 3c3 12368 ..^cfzo 13757 ♯chash 14442 ⟨“cs3 14961 Basecbs 17349 distcds 17399 0_gc0g 17572 Mgmcmgm 18776 TarskiGcstrkg 28823 Itvcitv 28829 LineGclng 28830 cgrAccgra 29248 AngMgmcangmgm 29307

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 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-oadd 8458 df-er 8695 df-ec 8697 df-qs 8701 df-map 8827 df-pm 8828 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-sup 9412 df-inf 9413 df-dju 9954 df-card 9992 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-nn 12306 df-2 12375 df-3 12376 df-4 12377 df-5 12378 df-6 12379 df-7 12380 df-8 12381 df-9 12382 df-n0 12577 df-xnn0 12650 df-z 12664 df-dec 12785 df-uz 12936 df-fz 13610 df-fzo 13758 df-hash 14443 df-word 14627 df-concat 14684 df-s1 14711 df-s2 14967 df-s3 14968 df-struct 17287 df-slot 17322 df-ndx 17334 df-base 17350 df-plusg 17403 df-mulr 17404 df-sca 17406 df-vsca 17407 df-ip 17408 df-tset 17409 df-ple 17410 df-ds 17412 df-0g 17574 df-imas 17642 df-qus 17643 df-mgm 18778 df-trkgc 28844 df-trkgb 28845 df-trkgcb 28846 df-trkgld 28848 df-trkg 28849 df-cgrg 28908 df-ismt 28930 df-leg 28980 df-hlg 28998 df-mir 29059 df-rag 29103 df-perpg 29105 df-hpg 29170 df-mid 29213 df-lmi 29214 df-cgra 29249 df-angmgm 29308

This theorem is used by: (None)

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