Theorem axcaucvglemcl 7962

Description: Lemma for axcaucvg 7967. Mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)

Hypotheses

Ref	Expression
axcaucvglemcl.n	⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
axcaucvglemcl.f	⊢ (𝜑 → 𝐹:𝑁⟶ℝ)

Assertion

Ref	Expression
axcaucvglemcl	⊢ ((𝜑 ∧ 𝐽 ∈ N) → (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)

Distinct variable groups:   𝑧,𝐹   𝐽,𝑙,𝑢,𝑧   𝑦,𝑙,𝑢   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑢,𝑙)   𝐹(𝑥,𝑦,𝑢,𝑙)   𝐽(𝑥,𝑦)   𝑁(𝑥,𝑦,𝑧,𝑢,𝑙)

Proof of Theorem axcaucvglemcl

Step	Hyp	Ref	Expression
1		pitonn 7915	. . . . . 6 ⊢ (𝐽 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
2		axcaucvglemcl.n	. . . . . 6 ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
3	1, 2	eleqtrrdi 2290	. . . . 5 ⊢ (𝐽 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
4		axcaucvglemcl.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑁⟶ℝ)
5	4	ffvelcdmda 5697	. . . . 5 ⊢ ((𝜑 ∧ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ)
6	3, 5	sylan2 286	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ)
7		elrealeu 7896	. . . 4 ⊢ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ ↔ ∃!𝑧 ∈ R ⟨𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
8	6, 7	sylib 122	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ∃!𝑧 ∈ R ⟨𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
9		eqcom 2198	. . . 4 ⊢ (⟨𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ↔ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
10	9	reubii 2683	. . 3 ⊢ (∃!𝑧 ∈ R ⟨𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ↔ ∃!𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
11	8, 10	sylib 122	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ∃!𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
12		riotacl 5892	. 2 ⊢ (∃!𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩ → (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)
13	11, 12	syl 14	1 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∃!wreu 2477  ⟨cop 3625  ∩ cint 3874   class class class wbr 4033  ⟶wf 5254  ‘cfv 5258  ℩crio 5876  (class class class)co 5922  1_oc1o 6467  [cec 6590  Ncnpi 7339   ~_Q ceq 7346   <_Q cltq 7352  1_Pc1p 7359   +_P cpp 7360   ~_R cer 7363  Rcnr 7364  0_Rc0r 7365  ℝcr 7878  1c1 7880   + caddc 7882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-i1p 7534  df-iplp 7535  df-enr 7793  df-nr 7794  df-plr 7795  df-0r 7798  df-1r 7799  df-c 7885  df-1 7887  df-r 7889  df-add 7890

This theorem is referenced by:  axcaucvglemf  7963  axcaucvglemval  7964