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Theorem nsgqusf1olem2 33422
Description: Lemma for nsgqusf1o 33424. (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 484 . . . . . . 7 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)))
2 nsgqusf1o.s . . . . . . . . . 10 𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}
32reqabi 3457 . . . . . . . . 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 33421 . . . . . . . . . . 11 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
1413anasss 466 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (SubGrp‘𝐺) ∧ 𝑁)) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
1514, 5eleqtrdi 2849 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (SubGrp‘𝐺) ∧ 𝑁)) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
163, 15sylan2b 594 . . . . . . . 8 ((𝜑𝑆) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
1716adantr 480 . . . . . . 7 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
181, 17eqeltrd 2839 . . . . . 6 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
1918r19.29an 3156 . . . . 5 ((𝜑 ∧ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
20 sseq2 4022 . . . . . . . 8 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑁𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}))
2112adantr 480 . . . . . . . . 9 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
22 simpr 484 . . . . . . . . 9 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ∈ (SubGrp‘𝑄))
234, 8, 9, 21, 22nsgmgclem 33419 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
245eleq2i 2831 . . . . . . . . 9 (𝑓𝑇𝑓 ∈ (SubGrp‘𝑄))
25 nsgsubg 19189 . . . . . . . . . . . . 13 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
2612, 25syl 17 . . . . . . . . . . . 12 (𝜑𝑁 ∈ (SubGrp‘𝐺))
274subgss 19158 . . . . . . . . . . . 12 (𝑁 ∈ (SubGrp‘𝐺) → 𝑁𝐵)
2826, 27syl 17 . . . . . . . . . . 11 (𝜑𝑁𝐵)
2928adantr 480 . . . . . . . . . 10 ((𝜑𝑓𝑇) → 𝑁𝐵)
3026ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
319grplsmid 33412 . . . . . . . . . . . 12 ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎𝑁) → ({𝑎} 𝑁) = 𝑁)
3230, 31sylancom 588 . . . . . . . . . . 11 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → ({𝑎} 𝑁) = 𝑁)
3324biimpi 216 . . . . . . . . . . . . 13 (𝑓𝑇𝑓 ∈ (SubGrp‘𝑄))
348nsgqus0 33418 . . . . . . . . . . . . 13 ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁𝑓)
3512, 33, 34syl2an 596 . . . . . . . . . . . 12 ((𝜑𝑓𝑇) → 𝑁𝑓)
3635adantr 480 . . . . . . . . . . 11 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → 𝑁𝑓)
3732, 36eqeltrd 2839 . . . . . . . . . 10 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → ({𝑎} 𝑁) ∈ 𝑓)
3829, 37ssrabdv 4084 . . . . . . . . 9 ((𝜑𝑓𝑇) → 𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
3924, 38sylan2br 595 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
4020, 23, 39elrabd 3697 . . . . . . 7 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ { ∈ (SubGrp‘𝐺) ∣ 𝑁})
4140, 2eleqtrrdi 2850 . . . . . 6 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ 𝑆)
42 mpteq1 5241 . . . . . . . . 9 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑥 ↦ ({𝑥} 𝑁)) = (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
4342rneqd 5952 . . . . . . . 8 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → ran (𝑥 ↦ ({𝑥} 𝑁)) = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
4443eqeq2d 2746 . . . . . . 7 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
4544adantl 481 . . . . . 6 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) → (𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
46 eqid 2735 . . . . . . . . . . . . . . 15 (Base‘𝑄) = (Base‘𝑄)
4746subgss 19158 . . . . . . . . . . . . . 14 (𝑓 ∈ (SubGrp‘𝑄) → 𝑓 ⊆ (Base‘𝑄))
4847adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ⊆ (Base‘𝑄))
4948sselda 3995 . . . . . . . . . . . 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 7466 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (𝐺 ~QG 𝑁) ∈ V)
53 subgrcl 19162 . . . . . . . . . . . . . . 15 (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5426, 53syl 17 . . . . . . . . . . . . . 14 (𝜑𝐺 ∈ Grp)
5554ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝐺 ∈ Grp)
5650, 51, 52, 55qusbas 17592 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
5749, 56eleqtrrd 2842 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)))
58 elqsi 8809 . . . . . . . . . . 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 4641 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → {𝑎} = {𝑥})
6160oveq1d 7446 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ({𝑎} 𝑁) = ({𝑥} 𝑁))
6261eleq1d 2824 . . . . . . . . . . . . . 14 (𝑎 = 𝑥 → (({𝑎} 𝑁) ∈ 𝑓 ↔ ({𝑥} 𝑁) ∈ 𝑓))
63 simplr 769 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥𝐵)
64 simpr 484 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
6526ad4antr 732 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
664, 9, 65, 63quslsm 33413 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} 𝑁))
6764, 66eqtrd 2775 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = ({𝑥} 𝑁))
68 simpllr 776 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖𝑓)
6967, 68eqeltrrd 2840 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → ({𝑥} 𝑁) ∈ 𝑓)
7062, 63, 69elrabd 3697 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
7170, 67jca 511 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} 𝑁)))
7271expl 457 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → ((𝑥𝐵𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} 𝑁))))
7372reximdv2 3162 . . . . . . . . . 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 769 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
7662elrab 3695 . . . . . . . . . . . 12 (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↔ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓))
7775, 76sylib 218 . . . . . . . . . . 11 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓))
78 simpllr 776 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → 𝑖 = ({𝑥} 𝑁))
79 simpr 484 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → ({𝑥} 𝑁) ∈ 𝑓)
8078, 79eqeltrd 2839 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → 𝑖𝑓)
8180anasss 466 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓)) → 𝑖𝑓)
8281adantllr 719 . . . . . . . . . . 11 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓)) → 𝑖𝑓)
8377, 82mpdan 687 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑖𝑓)
8483r19.29an 3156 . . . . . . . . 9 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)) → 𝑖𝑓)
8574, 84impbida 801 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → (𝑖𝑓 ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
86 eqid 2735 . . . . . . . . . 10 (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) = (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))
8786elrnmpt 5972 . . . . . . . . 9 (𝑖 ∈ V → (𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
8887elv 3483 . . . . . . . 8 (𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁))
8985, 88bitr4di 289 . . . . . . 7 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → (𝑖𝑓𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
9089eqrdv 2733 . . . . . 6 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
9141, 45, 90rspcedvd 3624 . . . . 5 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)))
9219, 91impbida 801 . . . 4 (𝜑 → (∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 ∈ (SubGrp‘𝑄)))
9392abbidv 2806 . . 3 (𝜑 → {𝑓 ∣ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))} = {𝑓𝑓 ∈ (SubGrp‘𝑄)})
9410rnmpt 5971 . . 3 ran 𝐸 = {𝑓 ∣ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))}
95 abid1 2876 . . 3 (SubGrp‘𝑄) = {𝑓𝑓 ∈ (SubGrp‘𝑄)}
9693, 94, 953eqtr4g 2800 . 2 (𝜑 → ran 𝐸 = (SubGrp‘𝑄))
9796, 5eqtr4di 2793 1 (𝜑 → ran 𝐸 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  {csn 4631  cmpt 5231  ran crn 5690  cfv 6563  (class class class)co 7431  [cec 8742   / cqs 8743  Basecbs 17245  lecple 17305   /s cqus 17552  toInccipo 18585  Grpcgrp 18964  SubGrpcsubg 19151  NrmSGrpcnsg 19152   ~QG cqg 19153  LSSumclsm 19667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-ec 8746  df-qs 8750  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17488  df-imas 17555  df-qus 17556  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-grp 18967  df-minusg 18968  df-subg 19154  df-nsg 19155  df-eqg 19156  df-oppg 19377  df-lsm 19669
This theorem is referenced by:  nsgqusf1o  33424
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