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Theorem cantnfp1 8461
 Description: If 𝐹 is created by adding a single term (𝐹‘𝑋) = 𝑌 to 𝐺, where 𝑋 is larger than any element of the support of 𝐺, then 𝐹 is also a finitely supported function and it is assigned the value ((𝐴 ↑𝑜 𝑋) ·𝑜 𝑌) +𝑜 𝑧 where 𝑧 is the value of 𝐺. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
cantnfp1.g (𝜑𝐺𝑆)
cantnfp1.x (𝜑𝑋𝐵)
cantnfp1.y (𝜑𝑌𝐴)
cantnfp1.s (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
cantnfp1.f 𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))
Assertion
Ref Expression
cantnfp1 (𝜑 → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
Distinct variable groups:   𝑡,𝐵   𝑡,𝐴   𝑡,𝑆   𝑡,𝐺   𝜑,𝑡   𝑡,𝑌   𝑡,𝑋
Allowed substitution hint:   𝐹(𝑡)

Proof of Theorem cantnfp1
Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfp1.f . . . . . 6 𝐹 = (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)))
2 cantnfs.b . . . . . . . . . . . . 13 (𝜑𝐵 ∈ On)
3 cantnfp1.x . . . . . . . . . . . . 13 (𝜑𝑋𝐵)
4 onelon 5665 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
52, 3, 4syl2anc 691 . . . . . . . . . . . 12 (𝜑𝑋 ∈ On)
6 eloni 5650 . . . . . . . . . . . 12 (𝑋 ∈ On → Ord 𝑋)
7 ordirr 5658 . . . . . . . . . . . 12 (Ord 𝑋 → ¬ 𝑋𝑋)
85, 6, 73syl 18 . . . . . . . . . . 11 (𝜑 → ¬ 𝑋𝑋)
9 fvex 6113 . . . . . . . . . . . . . 14 (𝐺𝑋) ∈ V
10 dif1o 7467 . . . . . . . . . . . . . 14 ((𝐺𝑋) ∈ (V ∖ 1𝑜) ↔ ((𝐺𝑋) ∈ V ∧ (𝐺𝑋) ≠ ∅))
119, 10mpbiran 955 . . . . . . . . . . . . 13 ((𝐺𝑋) ∈ (V ∖ 1𝑜) ↔ (𝐺𝑋) ≠ ∅)
12 cantnfp1.g . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐺𝑆)
13 cantnfs.s . . . . . . . . . . . . . . . . . . . . 21 𝑆 = dom (𝐴 CNF 𝐵)
14 cantnfs.a . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐴 ∈ On)
1513, 14, 2cantnfs 8446 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
1612, 15mpbid 221 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 474 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺:𝐵𝐴)
18 ffn 5958 . . . . . . . . . . . . . . . . . 18 (𝐺:𝐵𝐴𝐺 Fn 𝐵)
1917, 18syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 Fn 𝐵)
20 0ex 4718 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
2120a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ∅ ∈ V)
22 elsuppfn 7190 . . . . . . . . . . . . . . . . 17 ((𝐺 Fn 𝐵𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
2319, 2, 21, 22syl3anc 1318 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
2411bicomi 213 . . . . . . . . . . . . . . . . . 18 ((𝐺𝑋) ≠ ∅ ↔ (𝐺𝑋) ∈ (V ∖ 1𝑜))
2524a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐺𝑋) ≠ ∅ ↔ (𝐺𝑋) ∈ (V ∖ 1𝑜)))
2625anbi2d 736 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1𝑜))))
2723, 26bitrd 267 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1𝑜))))
28 cantnfp1.s . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
2928sseld 3567 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) → 𝑋𝑋))
3027, 29sylbird 249 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1𝑜)) → 𝑋𝑋))
313, 30mpand 707 . . . . . . . . . . . . 13 (𝜑 → ((𝐺𝑋) ∈ (V ∖ 1𝑜) → 𝑋𝑋))
3211, 31syl5bir 232 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑋) ≠ ∅ → 𝑋𝑋))
3332necon1bd 2800 . . . . . . . . . . 11 (𝜑 → (¬ 𝑋𝑋 → (𝐺𝑋) = ∅))
348, 33mpd 15 . . . . . . . . . 10 (𝜑 → (𝐺𝑋) = ∅)
3534ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → (𝐺𝑋) = ∅)
36 simpr 476 . . . . . . . . . 10 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑡 = 𝑋)
3736fveq2d 6107 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → (𝐺𝑡) = (𝐺𝑋))
38 simpllr 795 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = ∅)
3935, 37, 383eqtr4rd 2655 . . . . . . . 8 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = (𝐺𝑡))
40 eqidd 2611 . . . . . . . 8 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ ¬ 𝑡 = 𝑋) → (𝐺𝑡) = (𝐺𝑡))
4139, 40ifeqda 4071 . . . . . . 7 (((𝜑𝑌 = ∅) ∧ 𝑡𝐵) → if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)) = (𝐺𝑡))
4241mpteq2dva 4672 . . . . . 6 ((𝜑𝑌 = ∅) → (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡))) = (𝑡𝐵 ↦ (𝐺𝑡)))
431, 42syl5eq 2656 . . . . 5 ((𝜑𝑌 = ∅) → 𝐹 = (𝑡𝐵 ↦ (𝐺𝑡)))
4417feqmptd 6159 . . . . . 6 (𝜑𝐺 = (𝑡𝐵 ↦ (𝐺𝑡)))
4544adantr 480 . . . . 5 ((𝜑𝑌 = ∅) → 𝐺 = (𝑡𝐵 ↦ (𝐺𝑡)))
4643, 45eqtr4d 2647 . . . 4 ((𝜑𝑌 = ∅) → 𝐹 = 𝐺)
4712adantr 480 . . . 4 ((𝜑𝑌 = ∅) → 𝐺𝑆)
4846, 47eqeltrd 2688 . . 3 ((𝜑𝑌 = ∅) → 𝐹𝑆)
49 oecl 7504 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
5014, 2, 49syl2anc 691 . . . . . . 7 (𝜑 → (𝐴𝑜 𝐵) ∈ On)
5113, 14, 2cantnff 8454 . . . . . . . 8 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵))
5251, 12ffvelrnd 6268 . . . . . . 7 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴𝑜 𝐵))
53 onelon 5665 . . . . . . 7 (((𝐴𝑜 𝐵) ∈ On ∧ ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴𝑜 𝐵)) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
5450, 52, 53syl2anc 691 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
5554adantr 480 . . . . 5 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
56 oa0r 7505 . . . . 5 (((𝐴 CNF 𝐵)‘𝐺) ∈ On → (∅ +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = ((𝐴 CNF 𝐵)‘𝐺))
5755, 56syl 17 . . . 4 ((𝜑𝑌 = ∅) → (∅ +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = ((𝐴 CNF 𝐵)‘𝐺))
58 oveq2 6557 . . . . . 6 (𝑌 = ∅ → ((𝐴𝑜 𝑋) ·𝑜 𝑌) = ((𝐴𝑜 𝑋) ·𝑜 ∅))
59 oecl 7504 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
6014, 5, 59syl2anc 691 . . . . . . 7 (𝜑 → (𝐴𝑜 𝑋) ∈ On)
61 om0 7484 . . . . . . 7 ((𝐴𝑜 𝑋) ∈ On → ((𝐴𝑜 𝑋) ·𝑜 ∅) = ∅)
6260, 61syl 17 . . . . . 6 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 ∅) = ∅)
6358, 62sylan9eqr 2666 . . . . 5 ((𝜑𝑌 = ∅) → ((𝐴𝑜 𝑋) ·𝑜 𝑌) = ∅)
6463oveq1d 6564 . . . 4 ((𝜑𝑌 = ∅) → (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = (∅ +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
6546fveq2d 6107 . . . 4 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = ((𝐴 CNF 𝐵)‘𝐺))
6657, 64, 653eqtr4rd 2655 . . 3 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
6748, 66jca 553 . 2 ((𝜑𝑌 = ∅) → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
6814adantr 480 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐴 ∈ On)
692adantr 480 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐵 ∈ On)
7012adantr 480 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐺𝑆)
713adantr 480 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝑋𝐵)
72 cantnfp1.y . . . . 5 (𝜑𝑌𝐴)
7372adantr 480 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝑌𝐴)
7428adantr 480 . . . 4 ((𝜑𝑌 ≠ ∅) → (𝐺 supp ∅) ⊆ 𝑋)
7513, 68, 69, 70, 71, 73, 74, 1cantnfp1lem1 8458 . . 3 ((𝜑𝑌 ≠ ∅) → 𝐹𝑆)
76 onelon 5665 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
7714, 72, 76syl2anc 691 . . . . . 6 (𝜑𝑌 ∈ On)
78 on0eln0 5697 . . . . . 6 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
7977, 78syl 17 . . . . 5 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
8079biimpar 501 . . . 4 ((𝜑𝑌 ≠ ∅) → ∅ ∈ 𝑌)
81 eqid 2610 . . . 4 OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅))
82 eqid 2610 . . . 4 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , (𝐹 supp ∅))‘𝑘)) ·𝑜 (𝐹‘(OrdIso( E , (𝐹 supp ∅))‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , (𝐹 supp ∅))‘𝑘)) ·𝑜 (𝐹‘(OrdIso( E , (𝐹 supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)
83 eqid 2610 . . . 4 OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
84 eqid 2610 . . . 4 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·𝑜 (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·𝑜 (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)
8513, 68, 69, 70, 71, 73, 74, 1, 80, 81, 82, 83, 84cantnfp1lem3 8460 . . 3 ((𝜑𝑌 ≠ ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
8675, 85jca 553 . 2 ((𝜑𝑌 ≠ ∅) → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
8767, 86pm2.61dane 2869 1 (𝜑 → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643   E cep 4947  dom cdm 5038  Ord word 5639  Oncon0 5640   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   supp csupp 7182  seq𝜔cseqom 7429  1𝑜c1o 7440   +𝑜 coa 7444   ·𝑜 comu 7445   ↑𝑜 coe 7446   finSupp cfsupp 8158  OrdIsocoi 8297   CNF ccnf 8441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442 This theorem is referenced by:  cantnflem1d  8468  cantnflem1  8469  cantnflem3  8471
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