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Theorem om1val 25077
Description: The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
Hypotheses
Ref Expression
om1val.o 𝑂 = (𝐽 Ω1 𝑌)
om1val.b (𝜑𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
om1val.p (𝜑+ = (*𝑝𝐽))
om1val.k (𝜑𝐾 = (𝐽ko II))
om1val.j (𝜑𝐽 ∈ (TopOn‘𝑋))
om1val.y (𝜑𝑌𝑋)
Assertion
Ref Expression
om1val (𝜑𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
Distinct variable groups:   𝑓,𝐽   𝜑,𝑓   𝑓,𝑌
Allowed substitution hints:   𝐵(𝑓)   + (𝑓)   𝐾(𝑓)   𝑂(𝑓)   𝑋(𝑓)

Proof of Theorem om1val
Dummy variables 𝑦 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 om1val.o . 2 𝑂 = (𝐽 Ω1 𝑌)
2 df-om1 25053 . . . 4 Ω1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗ko II)⟩})
32a1i 11 . . 3 (𝜑 → Ω1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗ko II)⟩}))
4 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑗 = 𝐽)
54oveq2d 7447 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (II Cn 𝑗) = (II Cn 𝐽))
6 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑦 = 𝑌)
76eqeq2d 2746 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑓‘0) = 𝑦 ↔ (𝑓‘0) = 𝑌))
86eqeq2d 2746 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑌))
97, 8anbi12d 632 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)))
105, 9rabeqbidv 3452 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
11 om1val.b . . . . . . 7 (𝜑𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
1211adantr 480 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
1310, 12eqtr4d 2778 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = 𝐵)
1413opeq2d 4885 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩ = ⟨(Base‘ndx), 𝐵⟩)
154fveq2d 6911 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (*𝑝𝑗) = (*𝑝𝐽))
16 om1val.p . . . . . . 7 (𝜑+ = (*𝑝𝐽))
1716adantr 480 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → + = (*𝑝𝐽))
1815, 17eqtr4d 2778 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (*𝑝𝑗) = + )
1918opeq2d 4885 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(+g‘ndx), (*𝑝𝑗)⟩ = ⟨(+g‘ndx), + ⟩)
204oveq1d 7446 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗ko II) = (𝐽ko II))
21 om1val.k . . . . . . 7 (𝜑𝐾 = (𝐽ko II))
2221adantr 480 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝐾 = (𝐽ko II))
2320, 22eqtr4d 2778 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗ko II) = 𝐾)
2423opeq2d 4885 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(TopSet‘ndx), (𝑗ko II)⟩ = ⟨(TopSet‘ndx), 𝐾⟩)
2514, 19, 24tpeq123d 4753 . . 3 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗ko II)⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
26 unieq 4923 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2726adantl 481 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝐽)
28 om1val.j . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
29 toponuni 22936 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3028, 29syl 17 . . . . 5 (𝜑𝑋 = 𝐽)
3130adantr 480 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑋 = 𝐽)
3227, 31eqtr4d 2778 . . 3 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝑋)
33 topontop 22935 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3428, 33syl 17 . . 3 (𝜑𝐽 ∈ Top)
35 om1val.y . . 3 (𝜑𝑌𝑋)
36 tpex 7765 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩} ∈ V
3736a1i 11 . . 3 (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩} ∈ V)
383, 25, 32, 34, 35, 37ovmpodx 7584 . 2 (𝜑 → (𝐽 Ω1 𝑌) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
391, 38eqtrid 2787 1 (𝜑𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  {ctp 4635  cop 4637   cuni 4912  cfv 6563  (class class class)co 7431  cmpo 7433  0cc0 11153  1c1 11154  ndxcnx 17227  Basecbs 17245  +gcplusg 17298  TopSetcts 17304  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  ko cxko 23585  IIcii 24915  *𝑝cpco 25047   Ω1 comi 25048
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-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  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-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-topon 22933  df-om1 25053
This theorem is referenced by:  om1bas  25078  om1plusg  25081  om1tset  25082
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