Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nsgqusf1olem2 Structured version   Visualization version   GIF version

Theorem nsgqusf1olem2 31124
 Description: Lemma for nsgqusf1o 31126. (Contributed by Thierry Arnoux, 4-Aug-2024.)
Hypotheses
Ref Expression
nsgqusf1o.b 𝐵 = (Base‘𝐺)
nsgqusf1o.s 𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}
nsgqusf1o.t 𝑇 = (SubGrp‘𝑄)
nsgqusf1o.1 = (le‘(toInc‘𝑆))
nsgqusf1o.2 = (le‘(toInc‘𝑇))
nsgqusf1o.q 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
nsgqusf1o.p = (LSSum‘𝐺)
nsgqusf1o.e 𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))
nsgqusf1o.f 𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
nsgqusf1o.n (𝜑𝑁 ∈ (NrmSGrp‘𝐺))
Assertion
Ref Expression
nsgqusf1olem2 (𝜑 → ran 𝐸 = 𝑇)
Distinct variable groups:   ,𝑎,𝑓,,𝑥   𝐵,𝑎,𝑓,,𝑥   𝐸,𝑎,𝑓,,𝑥   𝑓,𝐹,,𝑥   𝐺,𝑎,𝑓,,𝑥   𝑁,𝑎,𝑓,,𝑥   𝑄,𝑎,𝑓,,𝑥   𝑆,𝑎,𝑓,,𝑥   𝑇,𝑎,𝑓,,𝑥   𝜑,𝑎,𝑓,,𝑥
Allowed substitution hints:   𝐹(𝑎)   (𝑥,𝑓,,𝑎)   (𝑥,𝑓,,𝑎)

Proof of Theorem nsgqusf1olem2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 simpr 488 . . . . . . 7 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)))
2 nsgqusf1o.s . . . . . . . . . 10 𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}
32rabeq2i 3400 . . . . . . . . 9 (𝑆 ↔ ( ∈ (SubGrp‘𝐺) ∧ 𝑁))
4 nsgqusf1o.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐺)
5 nsgqusf1o.t . . . . . . . . . . . 12 𝑇 = (SubGrp‘𝑄)
6 nsgqusf1o.1 . . . . . . . . . . . 12 = (le‘(toInc‘𝑆))
7 nsgqusf1o.2 . . . . . . . . . . . 12 = (le‘(toInc‘𝑇))
8 nsgqusf1o.q . . . . . . . . . . . 12 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
9 nsgqusf1o.p . . . . . . . . . . . 12 = (LSSum‘𝐺)
10 nsgqusf1o.e . . . . . . . . . . . 12 𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))
11 nsgqusf1o.f . . . . . . . . . . . 12 𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
12 nsgqusf1o.n . . . . . . . . . . . 12 (𝜑𝑁 ∈ (NrmSGrp‘𝐺))
134, 2, 5, 6, 7, 8, 9, 10, 11, 12nsgqusf1olem1 31123 . . . . . . . . . . 11 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
1413anasss 470 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (SubGrp‘𝐺) ∧ 𝑁)) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
1514, 5eleqtrdi 2862 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (SubGrp‘𝐺) ∧ 𝑁)) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
163, 15sylan2b 596 . . . . . . . 8 ((𝜑𝑆) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
1716adantr 484 . . . . . . 7 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
181, 17eqeltrd 2852 . . . . . 6 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
1918r19.29an 3212 . . . . 5 ((𝜑 ∧ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
20 sseq2 3920 . . . . . . . 8 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑁𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}))
2112adantr 484 . . . . . . . . 9 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
22 simpr 488 . . . . . . . . 9 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ∈ (SubGrp‘𝑄))
234, 8, 9, 21, 22nsgmgclem 31121 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
245eleq2i 2843 . . . . . . . . 9 (𝑓𝑇𝑓 ∈ (SubGrp‘𝑄))
25 nsgsubg 18382 . . . . . . . . . . . . 13 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
2612, 25syl 17 . . . . . . . . . . . 12 (𝜑𝑁 ∈ (SubGrp‘𝐺))
274subgss 18352 . . . . . . . . . . . 12 (𝑁 ∈ (SubGrp‘𝐺) → 𝑁𝐵)
2826, 27syl 17 . . . . . . . . . . 11 (𝜑𝑁𝐵)
2928adantr 484 . . . . . . . . . 10 ((𝜑𝑓𝑇) → 𝑁𝐵)
3026ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
319grplsmid 31117 . . . . . . . . . . . 12 ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎𝑁) → ({𝑎} 𝑁) = 𝑁)
3230, 31sylancom 591 . . . . . . . . . . 11 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → ({𝑎} 𝑁) = 𝑁)
3324biimpi 219 . . . . . . . . . . . . 13 (𝑓𝑇𝑓 ∈ (SubGrp‘𝑄))
348nsgqus0 31120 . . . . . . . . . . . . 13 ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁𝑓)
3512, 33, 34syl2an 598 . . . . . . . . . . . 12 ((𝜑𝑓𝑇) → 𝑁𝑓)
3635adantr 484 . . . . . . . . . . 11 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → 𝑁𝑓)
3732, 36eqeltrd 2852 . . . . . . . . . 10 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → ({𝑎} 𝑁) ∈ 𝑓)
3829, 37ssrabdv 3980 . . . . . . . . 9 ((𝜑𝑓𝑇) → 𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
3924, 38sylan2br 597 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
4020, 23, 39elrabd 3606 . . . . . . 7 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ { ∈ (SubGrp‘𝐺) ∣ 𝑁})
4140, 2eleqtrrdi 2863 . . . . . 6 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ 𝑆)
42 mpteq1 5123 . . . . . . . . 9 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑥 ↦ ({𝑥} 𝑁)) = (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
4342rneqd 5783 . . . . . . . 8 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → ran (𝑥 ↦ ({𝑥} 𝑁)) = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
4443eqeq2d 2769 . . . . . . 7 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
4544adantl 485 . . . . . 6 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) → (𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
46 eqid 2758 . . . . . . . . . . . . . . 15 (Base‘𝑄) = (Base‘𝑄)
4746subgss 18352 . . . . . . . . . . . . . 14 (𝑓 ∈ (SubGrp‘𝑄) → 𝑓 ⊆ (Base‘𝑄))
4847adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ⊆ (Base‘𝑄))
4948sselda 3894 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝑖 ∈ (Base‘𝑄))
508a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)))
514a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝐵 = (Base‘𝐺))
52 ovexd 7190 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (𝐺 ~QG 𝑁) ∈ V)
53 subgrcl 18356 . . . . . . . . . . . . . . 15 (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5426, 53syl 17 . . . . . . . . . . . . . 14 (𝜑𝐺 ∈ Grp)
5554ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝐺 ∈ Grp)
5650, 51, 52, 55qusbas 16881 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
5749, 56eleqtrrd 2855 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)))
58 elqsi 8365 . . . . . . . . . . 11 (𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)) → ∃𝑥𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁))
5957, 58syl 17 . . . . . . . . . 10 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → ∃𝑥𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁))
60 sneq 4535 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → {𝑎} = {𝑥})
6160oveq1d 7170 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ({𝑎} 𝑁) = ({𝑥} 𝑁))
6261eleq1d 2836 . . . . . . . . . . . . . 14 (𝑎 = 𝑥 → (({𝑎} 𝑁) ∈ 𝑓 ↔ ({𝑥} 𝑁) ∈ 𝑓))
63 simplr 768 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥𝐵)
64 simpr 488 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
6526ad4antr 731 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
664, 9, 65, 63quslsm 31118 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} 𝑁))
6764, 66eqtrd 2793 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = ({𝑥} 𝑁))
68 simpllr 775 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖𝑓)
6967, 68eqeltrrd 2853 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → ({𝑥} 𝑁) ∈ 𝑓)
7062, 63, 69elrabd 3606 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
7170, 67jca 515 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} 𝑁)))
7271expl 461 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → ((𝑥𝐵𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} 𝑁))))
7372reximdv2 3195 . . . . . . . . . 10 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (∃𝑥𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁) → ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
7459, 73mpd 15 . . . . . . . . 9 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁))
75 simplr 768 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
7662elrab 3604 . . . . . . . . . . . 12 (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↔ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓))
7775, 76sylib 221 . . . . . . . . . . 11 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓))
78 simpllr 775 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → 𝑖 = ({𝑥} 𝑁))
79 simpr 488 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → ({𝑥} 𝑁) ∈ 𝑓)
8078, 79eqeltrd 2852 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → 𝑖𝑓)
8180anasss 470 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓)) → 𝑖𝑓)
8281adantllr 718 . . . . . . . . . . 11 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓)) → 𝑖𝑓)
8377, 82mpdan 686 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑖𝑓)
8483r19.29an 3212 . . . . . . . . 9 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)) → 𝑖𝑓)
8574, 84impbida 800 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → (𝑖𝑓 ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
86 eqid 2758 . . . . . . . . . 10 (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) = (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))
8786elrnmpt 5801 . . . . . . . . 9 (𝑖 ∈ V → (𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
8887elv 3415 . . . . . . . 8 (𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁))
8985, 88bitr4di 292 . . . . . . 7 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → (𝑖𝑓𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
9089eqrdv 2756 . . . . . 6 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
9141, 45, 90rspcedvd 3546 . . . . 5 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)))
9219, 91impbida 800 . . . 4 (𝜑 → (∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 ∈ (SubGrp‘𝑄)))
9392abbidv 2822 . . 3 (𝜑 → {𝑓 ∣ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))} = {𝑓𝑓 ∈ (SubGrp‘𝑄)})
9410rnmpt 5800 . . 3 ran 𝐸 = {𝑓 ∣ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))}
95 abid1 2893 . . 3 (SubGrp‘𝑄) = {𝑓𝑓 ∈ (SubGrp‘𝑄)}
9693, 94, 953eqtr4g 2818 . 2 (𝜑 → ran 𝐸 = (SubGrp‘𝑄))
9796, 5eqtr4di 2811 1 (𝜑 → ran 𝐸 = 𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3860  {csn 4525   ↦ cmpt 5115  ran crn 5528  ‘cfv 6339  (class class class)co 7155  [cec 8302   / cqs 8303  Basecbs 16546  lecple 16635   /s cqus 16841  toInccipo 17832  Grpcgrp 18174  SubGrpcsubg 18345  NrmSGrpcnsg 18346   ~QG cqg 18347  LSSumclsm 18831 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-tpos 7907  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-er 8304  df-ec 8306  df-qs 8310  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-sup 8944  df-inf 8945  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-2 11742  df-3 11743  df-4 11744  df-5 11745  df-6 11746  df-7 11747  df-8 11748  df-9 11749  df-n0 11940  df-z 12026  df-dec 12143  df-uz 12288  df-fz 12945  df-struct 16548  df-ndx 16549  df-slot 16550  df-base 16552  df-sets 16553  df-ress 16554  df-plusg 16641  df-mulr 16642  df-sca 16644  df-vsca 16645  df-ip 16646  df-tset 16647  df-ple 16648  df-ds 16650  df-0g 16778  df-imas 16844  df-qus 16845  df-mgm 17923  df-sgrp 17972  df-mnd 17983  df-submnd 18028  df-grp 18177  df-minusg 18178  df-subg 18348  df-nsg 18349  df-eqg 18350  df-oppg 18546  df-lsm 18833 This theorem is referenced by:  nsgqusf1o  31126
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