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Theorem cantnfp1 8538
Description: If 𝐹 is created by adding a single term (𝐹𝑋) = 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 ((𝐴𝑜 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 5717 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
52, 3, 4syl2anc 692 . . . . . . . . . . . 12 (𝜑𝑋 ∈ On)
6 eloni 5702 . . . . . . . . . . . 12 (𝑋 ∈ On → Ord 𝑋)
7 ordirr 5710 . . . . . . . . . . . 12 (Ord 𝑋 → ¬ 𝑋𝑋)
85, 6, 73syl 18 . . . . . . . . . . 11 (𝜑 → ¬ 𝑋𝑋)
9 fvex 6168 . . . . . . . . . . . . . 14 (𝐺𝑋) ∈ V
10 dif1o 7540 . . . . . . . . . . . . . 14 ((𝐺𝑋) ∈ (V ∖ 1𝑜) ↔ ((𝐺𝑋) ∈ V ∧ (𝐺𝑋) ≠ ∅))
119, 10mpbiran 952 . . . . . . . . . . . . 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 8523 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
1612, 15mpbid 222 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 475 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺:𝐵𝐴)
18 ffn 6012 . . . . . . . . . . . . . . . . . 18 (𝐺:𝐵𝐴𝐺 Fn 𝐵)
1917, 18syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 Fn 𝐵)
20 0ex 4760 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
2120a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ∅ ∈ V)
22 elsuppfn 7263 . . . . . . . . . . . . . . . . 17 ((𝐺 Fn 𝐵𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
2319, 2, 21, 22syl3anc 1323 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
2411bicomi 214 . . . . . . . . . . . . . . . . . 18 ((𝐺𝑋) ≠ ∅ ↔ (𝐺𝑋) ∈ (V ∖ 1𝑜))
2524a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐺𝑋) ≠ ∅ ↔ (𝐺𝑋) ∈ (V ∖ 1𝑜)))
2625anbi2d 739 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1𝑜))))
2723, 26bitrd 268 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1𝑜))))
28 cantnfp1.s . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
2928sseld 3587 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) → 𝑋𝑋))
3027, 29sylbird 250 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1𝑜)) → 𝑋𝑋))
313, 30mpand 710 . . . . . . . . . . . . 13 (𝜑 → ((𝐺𝑋) ∈ (V ∖ 1𝑜) → 𝑋𝑋))
3211, 31syl5bir 233 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑋) ≠ ∅ → 𝑋𝑋))
3332necon1bd 2808 . . . . . . . . . . 11 (𝜑 → (¬ 𝑋𝑋 → (𝐺𝑋) = ∅))
348, 33mpd 15 . . . . . . . . . 10 (𝜑 → (𝐺𝑋) = ∅)
3534ad3antrrr 765 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → (𝐺𝑋) = ∅)
36 simpr 477 . . . . . . . . . 10 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑡 = 𝑋)
3736fveq2d 6162 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → (𝐺𝑡) = (𝐺𝑋))
38 simpllr 798 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = ∅)
3935, 37, 383eqtr4rd 2666 . . . . . . . 8 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = (𝐺𝑡))
40 eqidd 2622 . . . . . . . 8 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ ¬ 𝑡 = 𝑋) → (𝐺𝑡) = (𝐺𝑡))
4139, 40ifeqda 4099 . . . . . . 7 (((𝜑𝑌 = ∅) ∧ 𝑡𝐵) → if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)) = (𝐺𝑡))
4241mpteq2dva 4714 . . . . . 6 ((𝜑𝑌 = ∅) → (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡))) = (𝑡𝐵 ↦ (𝐺𝑡)))
431, 42syl5eq 2667 . . . . 5 ((𝜑𝑌 = ∅) → 𝐹 = (𝑡𝐵 ↦ (𝐺𝑡)))
4417feqmptd 6216 . . . . . 6 (𝜑𝐺 = (𝑡𝐵 ↦ (𝐺𝑡)))
4544adantr 481 . . . . 5 ((𝜑𝑌 = ∅) → 𝐺 = (𝑡𝐵 ↦ (𝐺𝑡)))
4643, 45eqtr4d 2658 . . . 4 ((𝜑𝑌 = ∅) → 𝐹 = 𝐺)
4712adantr 481 . . . 4 ((𝜑𝑌 = ∅) → 𝐺𝑆)
4846, 47eqeltrd 2698 . . 3 ((𝜑𝑌 = ∅) → 𝐹𝑆)
49 oecl 7577 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
5014, 2, 49syl2anc 692 . . . . . . 7 (𝜑 → (𝐴𝑜 𝐵) ∈ On)
5113, 14, 2cantnff 8531 . . . . . . . 8 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵))
5251, 12ffvelrnd 6326 . . . . . . 7 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴𝑜 𝐵))
53 onelon 5717 . . . . . . 7 (((𝐴𝑜 𝐵) ∈ On ∧ ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴𝑜 𝐵)) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
5450, 52, 53syl2anc 692 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
5554adantr 481 . . . . 5 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
56 oa0r 7578 . . . . 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 6623 . . . . . 6 (𝑌 = ∅ → ((𝐴𝑜 𝑋) ·𝑜 𝑌) = ((𝐴𝑜 𝑋) ·𝑜 ∅))
59 oecl 7577 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
6014, 5, 59syl2anc 692 . . . . . . 7 (𝜑 → (𝐴𝑜 𝑋) ∈ On)
61 om0 7557 . . . . . . 7 ((𝐴𝑜 𝑋) ∈ On → ((𝐴𝑜 𝑋) ·𝑜 ∅) = ∅)
6260, 61syl 17 . . . . . 6 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 ∅) = ∅)
6358, 62sylan9eqr 2677 . . . . 5 ((𝜑𝑌 = ∅) → ((𝐴𝑜 𝑋) ·𝑜 𝑌) = ∅)
6463oveq1d 6630 . . . 4 ((𝜑𝑌 = ∅) → (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = (∅ +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
6546fveq2d 6162 . . . 4 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = ((𝐴 CNF 𝐵)‘𝐺))
6657, 64, 653eqtr4rd 2666 . . 3 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
6748, 66jca 554 . 2 ((𝜑𝑌 = ∅) → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
6814adantr 481 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐴 ∈ On)
692adantr 481 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐵 ∈ On)
7012adantr 481 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐺𝑆)
713adantr 481 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝑋𝐵)
72 cantnfp1.y . . . . 5 (𝜑𝑌𝐴)
7372adantr 481 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝑌𝐴)
7428adantr 481 . . . 4 ((𝜑𝑌 ≠ ∅) → (𝐺 supp ∅) ⊆ 𝑋)
7513, 68, 69, 70, 71, 73, 74, 1cantnfp1lem1 8535 . . 3 ((𝜑𝑌 ≠ ∅) → 𝐹𝑆)
76 onelon 5717 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
7714, 72, 76syl2anc 692 . . . . . 6 (𝜑𝑌 ∈ On)
78 on0eln0 5749 . . . . . 6 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
7977, 78syl 17 . . . . 5 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
8079biimpar 502 . . . 4 ((𝜑𝑌 ≠ ∅) → ∅ ∈ 𝑌)
81 eqid 2621 . . . 4 OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅))
82 eqid 2621 . . . 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 2621 . . . 4 OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
84 eqid 2621 . . . 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 8537 . . 3 ((𝜑𝑌 ≠ ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
8675, 85jca 554 . 2 ((𝜑𝑌 ≠ ∅) → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
8767, 86pm2.61dane 2877 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 196  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3190  cdif 3557  wss 3560  c0 3897  ifcif 4064   class class class wbr 4623  cmpt 4683   E cep 4993  dom cdm 5084  Ord word 5691  Oncon0 5692   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  cmpt2 6617   supp csupp 7255  seq𝜔cseqom 7502  1𝑜c1o 7513   +𝑜 coa 7517   ·𝑜 comu 7518  𝑜 coe 7519   finSupp cfsupp 8235  OrdIsocoi 8374   CNF ccnf 8518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-seqom 7503  df-1o 7520  df-2o 7521  df-oadd 7524  df-omul 7525  df-oexp 7526  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-oi 8375  df-cnf 8519
This theorem is referenced by:  cantnflem1d  8545  cantnflem1  8546  cantnflem3  8548
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