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Theorem nsgqusf1olem2 32192
Description: Lemma for nsgqusf1o 32194. (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 485 . . . . . . 7 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)))
2 nsgqusf1o.s . . . . . . . . . 10 𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}
32reqabi 3429 . . . . . . . . 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 32191 . . . . . . . . . . 11 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
1413anasss 467 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (SubGrp‘𝐺) ∧ 𝑁)) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
1514, 5eleqtrdi 2848 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (SubGrp‘𝐺) ∧ 𝑁)) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
163, 15sylan2b 594 . . . . . . . 8 ((𝜑𝑆) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
1716adantr 481 . . . . . . 7 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
181, 17eqeltrd 2838 . . . . . 6 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
1918r19.29an 3155 . . . . 5 ((𝜑 ∧ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
20 sseq2 3970 . . . . . . . 8 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑁𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}))
2112adantr 481 . . . . . . . . 9 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
22 simpr 485 . . . . . . . . 9 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ∈ (SubGrp‘𝑄))
234, 8, 9, 21, 22nsgmgclem 32189 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
245eleq2i 2829 . . . . . . . . 9 (𝑓𝑇𝑓 ∈ (SubGrp‘𝑄))
25 nsgsubg 18960 . . . . . . . . . . . . 13 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
2612, 25syl 17 . . . . . . . . . . . 12 (𝜑𝑁 ∈ (SubGrp‘𝐺))
274subgss 18929 . . . . . . . . . . . 12 (𝑁 ∈ (SubGrp‘𝐺) → 𝑁𝐵)
2826, 27syl 17 . . . . . . . . . . 11 (𝜑𝑁𝐵)
2928adantr 481 . . . . . . . . . 10 ((𝜑𝑓𝑇) → 𝑁𝐵)
3026ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
319grplsmid 32185 . . . . . . . . . . . 12 ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎𝑁) → ({𝑎} 𝑁) = 𝑁)
3230, 31sylancom 588 . . . . . . . . . . 11 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → ({𝑎} 𝑁) = 𝑁)
3324biimpi 215 . . . . . . . . . . . . 13 (𝑓𝑇𝑓 ∈ (SubGrp‘𝑄))
348nsgqus0 32188 . . . . . . . . . . . . 13 ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁𝑓)
3512, 33, 34syl2an 596 . . . . . . . . . . . 12 ((𝜑𝑓𝑇) → 𝑁𝑓)
3635adantr 481 . . . . . . . . . . 11 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → 𝑁𝑓)
3732, 36eqeltrd 2838 . . . . . . . . . 10 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → ({𝑎} 𝑁) ∈ 𝑓)
3829, 37ssrabdv 4031 . . . . . . . . 9 ((𝜑𝑓𝑇) → 𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
3924, 38sylan2br 595 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
4020, 23, 39elrabd 3647 . . . . . . 7 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ { ∈ (SubGrp‘𝐺) ∣ 𝑁})
4140, 2eleqtrrdi 2849 . . . . . 6 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ 𝑆)
42 mpteq1 5198 . . . . . . . . 9 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑥 ↦ ({𝑥} 𝑁)) = (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
4342rneqd 5893 . . . . . . . 8 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → ran (𝑥 ↦ ({𝑥} 𝑁)) = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
4443eqeq2d 2747 . . . . . . 7 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
4544adantl 482 . . . . . 6 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) → (𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
46 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘𝑄) = (Base‘𝑄)
4746subgss 18929 . . . . . . . . . . . . . 14 (𝑓 ∈ (SubGrp‘𝑄) → 𝑓 ⊆ (Base‘𝑄))
4847adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ⊆ (Base‘𝑄))
4948sselda 3944 . . . . . . . . . . . 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 7392 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (𝐺 ~QG 𝑁) ∈ V)
53 subgrcl 18933 . . . . . . . . . . . . . . 15 (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5426, 53syl 17 . . . . . . . . . . . . . 14 (𝜑𝐺 ∈ Grp)
5554ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝐺 ∈ Grp)
5650, 51, 52, 55qusbas 17427 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
5749, 56eleqtrrd 2841 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)))
58 elqsi 8709 . . . . . . . . . . 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 4596 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → {𝑎} = {𝑥})
6160oveq1d 7372 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ({𝑎} 𝑁) = ({𝑥} 𝑁))
6261eleq1d 2822 . . . . . . . . . . . . . 14 (𝑎 = 𝑥 → (({𝑎} 𝑁) ∈ 𝑓 ↔ ({𝑥} 𝑁) ∈ 𝑓))
63 simplr 767 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥𝐵)
64 simpr 485 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
6526ad4antr 730 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
664, 9, 65, 63quslsm 32186 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} 𝑁))
6764, 66eqtrd 2776 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = ({𝑥} 𝑁))
68 simpllr 774 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖𝑓)
6967, 68eqeltrrd 2839 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → ({𝑥} 𝑁) ∈ 𝑓)
7062, 63, 69elrabd 3647 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
7170, 67jca 512 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} 𝑁)))
7271expl 458 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → ((𝑥𝐵𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} 𝑁))))
7372reximdv2 3161 . . . . . . . . . 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 767 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
7662elrab 3645 . . . . . . . . . . . 12 (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↔ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓))
7775, 76sylib 217 . . . . . . . . . . 11 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓))
78 simpllr 774 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → 𝑖 = ({𝑥} 𝑁))
79 simpr 485 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → ({𝑥} 𝑁) ∈ 𝑓)
8078, 79eqeltrd 2838 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → 𝑖𝑓)
8180anasss 467 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓)) → 𝑖𝑓)
8281adantllr 717 . . . . . . . . . . 11 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓)) → 𝑖𝑓)
8377, 82mpdan 685 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑖𝑓)
8483r19.29an 3155 . . . . . . . . 9 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)) → 𝑖𝑓)
8574, 84impbida 799 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → (𝑖𝑓 ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
86 eqid 2736 . . . . . . . . . 10 (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) = (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))
8786elrnmpt 5911 . . . . . . . . 9 (𝑖 ∈ V → (𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
8887elv 3451 . . . . . . . 8 (𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁))
8985, 88bitr4di 288 . . . . . . 7 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → (𝑖𝑓𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
9089eqrdv 2734 . . . . . 6 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
9141, 45, 90rspcedvd 3583 . . . . 5 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)))
9219, 91impbida 799 . . . 4 (𝜑 → (∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 ∈ (SubGrp‘𝑄)))
9392abbidv 2805 . . 3 (𝜑 → {𝑓 ∣ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))} = {𝑓𝑓 ∈ (SubGrp‘𝑄)})
9410rnmpt 5910 . . 3 ran 𝐸 = {𝑓 ∣ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))}
95 abid1 2874 . . 3 (SubGrp‘𝑄) = {𝑓𝑓 ∈ (SubGrp‘𝑄)}
9693, 94, 953eqtr4g 2801 . 2 (𝜑 → ran 𝐸 = (SubGrp‘𝑄))
9796, 5eqtr4di 2794 1 (𝜑 → ran 𝐸 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  {csn 4586  cmpt 5188  ran crn 5634  cfv 6496  (class class class)co 7357  [cec 8646   / cqs 8647  Basecbs 17083  lecple 17140   /s cqus 17387  toInccipo 18416  Grpcgrp 18748  SubGrpcsubg 18922  NrmSGrpcnsg 18923   ~QG cqg 18924  LSSumclsm 19416
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-ec 8650  df-qs 8654  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-0g 17323  df-imas 17390  df-qus 17391  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-subg 18925  df-nsg 18926  df-eqg 18927  df-oppg 19124  df-lsm 19418
This theorem is referenced by:  nsgqusf1o  32194
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