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Theorem nsgqusf1olem2 33442
Description: Lemma for nsgqusf1o 33444. (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 3460 . . . . . . . . 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 33441 . . . . . . . . . . 11 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
1413anasss 466 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (SubGrp‘𝐺) ∧ 𝑁)) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
1514, 5eleqtrdi 2851 . . . . . . . . 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 2841 . . . . . 6 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
1918r19.29an 3158 . . . . 5 ((𝜑 ∧ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
20 sseq2 4010 . . . . . . . 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 33439 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
245eleq2i 2833 . . . . . . . . 9 (𝑓𝑇𝑓 ∈ (SubGrp‘𝑄))
25 nsgsubg 19176 . . . . . . . . . . . . 13 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
2612, 25syl 17 . . . . . . . . . . . 12 (𝜑𝑁 ∈ (SubGrp‘𝐺))
274subgss 19145 . . . . . . . . . . . 12 (𝑁 ∈ (SubGrp‘𝐺) → 𝑁𝐵)
2826, 27syl 17 . . . . . . . . . . 11 (𝜑𝑁𝐵)
2928adantr 480 . . . . . . . . . 10 ((𝜑𝑓𝑇) → 𝑁𝐵)
3026ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
319grplsmid 33432 . . . . . . . . . . . 12 ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎𝑁) → ({𝑎} 𝑁) = 𝑁)
3230, 31sylancom 588 . . . . . . . . . . 11 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → ({𝑎} 𝑁) = 𝑁)
3324biimpi 216 . . . . . . . . . . . . 13 (𝑓𝑇𝑓 ∈ (SubGrp‘𝑄))
348nsgqus0 33438 . . . . . . . . . . . . 13 ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁𝑓)
3512, 33, 34syl2an 596 . . . . . . . . . . . 12 ((𝜑𝑓𝑇) → 𝑁𝑓)
3635adantr 480 . . . . . . . . . . 11 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → 𝑁𝑓)
3732, 36eqeltrd 2841 . . . . . . . . . 10 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → ({𝑎} 𝑁) ∈ 𝑓)
3829, 37ssrabdv 4074 . . . . . . . . 9 ((𝜑𝑓𝑇) → 𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
3924, 38sylan2br 595 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
4020, 23, 39elrabd 3694 . . . . . . 7 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ { ∈ (SubGrp‘𝐺) ∣ 𝑁})
4140, 2eleqtrrdi 2852 . . . . . 6 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ 𝑆)
42 mpteq1 5235 . . . . . . . . 9 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑥 ↦ ({𝑥} 𝑁)) = (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
4342rneqd 5949 . . . . . . . 8 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → ran (𝑥 ↦ ({𝑥} 𝑁)) = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
4443eqeq2d 2748 . . . . . . 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 2737 . . . . . . . . . . . . . . 15 (Base‘𝑄) = (Base‘𝑄)
4746subgss 19145 . . . . . . . . . . . . . 14 (𝑓 ∈ (SubGrp‘𝑄) → 𝑓 ⊆ (Base‘𝑄))
4847adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ⊆ (Base‘𝑄))
4948sselda 3983 . . . . . . . . . . . 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 19149 . . . . . . . . . . . . . . 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 17590 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
5749, 56eleqtrrd 2844 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)))
58 elqsi 8810 . . . . . . . . . . 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 4636 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → {𝑎} = {𝑥})
6160oveq1d 7446 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ({𝑎} 𝑁) = ({𝑥} 𝑁))
6261eleq1d 2826 . . . . . . . . . . . . . 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 33433 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} 𝑁))
6764, 66eqtrd 2777 . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → ({𝑥} 𝑁) ∈ 𝑓)
7062, 63, 69elrabd 3694 . . . . . . . . . . . . 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 3164 . . . . . . . . . 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 3692 . . . . . . . . . . . 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 2841 . . . . . . . . . . . . 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 3158 . . . . . . . . 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 2737 . . . . . . . . . 10 (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) = (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))
8786elrnmpt 5969 . . . . . . . . 9 (𝑖 ∈ V → (𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
8887elv 3485 . . . . . . . 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 2735 . . . . . 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 2808 . . 3 (𝜑 → {𝑓 ∣ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))} = {𝑓𝑓 ∈ (SubGrp‘𝑄)})
9410rnmpt 5968 . . 3 ran 𝐸 = {𝑓 ∣ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))}
95 abid1 2878 . . 3 (SubGrp‘𝑄) = {𝑓𝑓 ∈ (SubGrp‘𝑄)}
9693, 94, 953eqtr4g 2802 . 2 (𝜑 → ran 𝐸 = (SubGrp‘𝑄))
9796, 5eqtr4di 2795 1 (𝜑 → ran 𝐸 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  {csn 4626  cmpt 5225  ran crn 5686  cfv 6561  (class class class)co 7431  [cec 8743   / cqs 8744  Basecbs 17247  lecple 17304   /s cqus 17550  toInccipo 18572  Grpcgrp 18951  SubGrpcsubg 19138  NrmSGrpcnsg 19139   ~QG cqg 19140  LSSumclsm 19652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-ec 8747  df-qs 8751  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-0g 17486  df-imas 17553  df-qus 17554  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-subg 19141  df-nsg 19142  df-eqg 19143  df-oppg 19364  df-lsm 19654
This theorem is referenced by:  nsgqusf1o  33444
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