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Theorem cantnfp1 8822
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 5958 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
52, 3, 4syl2anc 575 . . . . . . . . . . . 12 (𝜑𝑋 ∈ On)
6 eloni 5943 . . . . . . . . . . . 12 (𝑋 ∈ On → Ord 𝑋)
7 ordirr 5951 . . . . . . . . . . . 12 (Ord 𝑋 → ¬ 𝑋𝑋)
85, 6, 73syl 18 . . . . . . . . . . 11 (𝜑 → ¬ 𝑋𝑋)
9 fvex 6418 . . . . . . . . . . . . . 14 (𝐺𝑋) ∈ V
10 dif1o 7814 . . . . . . . . . . . . . 14 ((𝐺𝑋) ∈ (V ∖ 1𝑜) ↔ ((𝐺𝑋) ∈ V ∧ (𝐺𝑋) ≠ ∅))
119, 10mpbiran 691 . . . . . . . . . . . . 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 8807 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
1612, 15mpbid 223 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 484 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺:𝐵𝐴)
1817ffnd 6254 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 Fn 𝐵)
19 0ex 4981 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
2019a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ∅ ∈ V)
21 elsuppfn 7534 . . . . . . . . . . . . . . . . 17 ((𝐺 Fn 𝐵𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
2218, 2, 20, 21syl3anc 1483 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
2311bicomi 215 . . . . . . . . . . . . . . . . . 18 ((𝐺𝑋) ≠ ∅ ↔ (𝐺𝑋) ∈ (V ∖ 1𝑜))
2423a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐺𝑋) ≠ ∅ ↔ (𝐺𝑋) ∈ (V ∖ 1𝑜)))
2524anbi2d 616 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1𝑜))))
2622, 25bitrd 270 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1𝑜))))
27 cantnfp1.s . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
2827sseld 3794 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) → 𝑋𝑋))
2926, 28sylbird 251 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1𝑜)) → 𝑋𝑋))
303, 29mpand 678 . . . . . . . . . . . . 13 (𝜑 → ((𝐺𝑋) ∈ (V ∖ 1𝑜) → 𝑋𝑋))
3111, 30syl5bir 234 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑋) ≠ ∅ → 𝑋𝑋))
3231necon1bd 2995 . . . . . . . . . . 11 (𝜑 → (¬ 𝑋𝑋 → (𝐺𝑋) = ∅))
338, 32mpd 15 . . . . . . . . . 10 (𝜑 → (𝐺𝑋) = ∅)
3433ad3antrrr 712 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → (𝐺𝑋) = ∅)
35 simpr 473 . . . . . . . . . 10 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑡 = 𝑋)
3635fveq2d 6409 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → (𝐺𝑡) = (𝐺𝑋))
37 simpllr 784 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = ∅)
3834, 36, 373eqtr4rd 2850 . . . . . . . 8 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = (𝐺𝑡))
39 eqidd 2806 . . . . . . . 8 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ ¬ 𝑡 = 𝑋) → (𝐺𝑡) = (𝐺𝑡))
4038, 39ifeqda 4311 . . . . . . 7 (((𝜑𝑌 = ∅) ∧ 𝑡𝐵) → if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)) = (𝐺𝑡))
4140mpteq2dva 4934 . . . . . 6 ((𝜑𝑌 = ∅) → (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡))) = (𝑡𝐵 ↦ (𝐺𝑡)))
421, 41syl5eq 2851 . . . . 5 ((𝜑𝑌 = ∅) → 𝐹 = (𝑡𝐵 ↦ (𝐺𝑡)))
4317feqmptd 6467 . . . . . 6 (𝜑𝐺 = (𝑡𝐵 ↦ (𝐺𝑡)))
4443adantr 468 . . . . 5 ((𝜑𝑌 = ∅) → 𝐺 = (𝑡𝐵 ↦ (𝐺𝑡)))
4542, 44eqtr4d 2842 . . . 4 ((𝜑𝑌 = ∅) → 𝐹 = 𝐺)
4612adantr 468 . . . 4 ((𝜑𝑌 = ∅) → 𝐺𝑆)
4745, 46eqeltrd 2884 . . 3 ((𝜑𝑌 = ∅) → 𝐹𝑆)
48 oecl 7851 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
4914, 2, 48syl2anc 575 . . . . . . 7 (𝜑 → (𝐴𝑜 𝐵) ∈ On)
5013, 14, 2cantnff 8815 . . . . . . . 8 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴𝑜 𝐵))
5150, 12ffvelrnd 6579 . . . . . . 7 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴𝑜 𝐵))
52 onelon 5958 . . . . . . 7 (((𝐴𝑜 𝐵) ∈ On ∧ ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴𝑜 𝐵)) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
5349, 51, 52syl2anc 575 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
5453adantr 468 . . . . 5 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
55 oa0r 7852 . . . . 5 (((𝐴 CNF 𝐵)‘𝐺) ∈ On → (∅ +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = ((𝐴 CNF 𝐵)‘𝐺))
5654, 55syl 17 . . . 4 ((𝜑𝑌 = ∅) → (∅ +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = ((𝐴 CNF 𝐵)‘𝐺))
57 oveq2 6879 . . . . . 6 (𝑌 = ∅ → ((𝐴𝑜 𝑋) ·𝑜 𝑌) = ((𝐴𝑜 𝑋) ·𝑜 ∅))
58 oecl 7851 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴𝑜 𝑋) ∈ On)
5914, 5, 58syl2anc 575 . . . . . . 7 (𝜑 → (𝐴𝑜 𝑋) ∈ On)
60 om0 7831 . . . . . . 7 ((𝐴𝑜 𝑋) ∈ On → ((𝐴𝑜 𝑋) ·𝑜 ∅) = ∅)
6159, 60syl 17 . . . . . 6 (𝜑 → ((𝐴𝑜 𝑋) ·𝑜 ∅) = ∅)
6257, 61sylan9eqr 2861 . . . . 5 ((𝜑𝑌 = ∅) → ((𝐴𝑜 𝑋) ·𝑜 𝑌) = ∅)
6362oveq1d 6886 . . . 4 ((𝜑𝑌 = ∅) → (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = (∅ +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
6445fveq2d 6409 . . . 4 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = ((𝐴 CNF 𝐵)‘𝐺))
6556, 63, 643eqtr4rd 2850 . . 3 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
6647, 65jca 503 . 2 ((𝜑𝑌 = ∅) → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
6714adantr 468 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐴 ∈ On)
682adantr 468 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐵 ∈ On)
6912adantr 468 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐺𝑆)
703adantr 468 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝑋𝐵)
71 cantnfp1.y . . . . 5 (𝜑𝑌𝐴)
7271adantr 468 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝑌𝐴)
7327adantr 468 . . . 4 ((𝜑𝑌 ≠ ∅) → (𝐺 supp ∅) ⊆ 𝑋)
7413, 67, 68, 69, 70, 72, 73, 1cantnfp1lem1 8819 . . 3 ((𝜑𝑌 ≠ ∅) → 𝐹𝑆)
75 onelon 5958 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
7614, 71, 75syl2anc 575 . . . . . 6 (𝜑𝑌 ∈ On)
77 on0eln0 5990 . . . . . 6 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
7876, 77syl 17 . . . . 5 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
7978biimpar 465 . . . 4 ((𝜑𝑌 ≠ ∅) → ∅ ∈ 𝑌)
80 eqid 2805 . . . 4 OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅))
81 eqid 2805 . . . 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)), ∅)
82 eqid 2805 . . . 4 OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
83 eqid 2805 . . . 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)), ∅)
8413, 67, 68, 69, 70, 72, 73, 1, 79, 80, 81, 82, 83cantnfp1lem3 8821 . . 3 ((𝜑𝑌 ≠ ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
8574, 84jca 503 . 2 ((𝜑𝑌 ≠ ∅) → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
8666, 85pm2.61dane 3064 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 197  wa 384   = wceq 1637  wcel 2158  wne 2977  Vcvv 3390  cdif 3763  wss 3766  c0 4113  ifcif 4276   class class class wbr 4840  cmpt 4919   E cep 5220  dom cdm 5308  Ord word 5932  Oncon0 5933   Fn wfn 6093  wf 6094  cfv 6098  (class class class)co 6871  cmpt2 6873   supp csupp 7526  seq𝜔cseqom 7775  1𝑜c1o 7786   +𝑜 coa 7790   ·𝑜 comu 7791  𝑜 coe 7792   finSupp cfsupp 8511  OrdIsocoi 8650   CNF ccnf 8802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-reu 3102  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-se 5268  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-isom 6107  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-1st 7395  df-2nd 7396  df-supp 7527  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-seqom 7776  df-1o 7793  df-2o 7794  df-oadd 7797  df-omul 7798  df-oexp 7799  df-er 7976  df-map 8091  df-en 8190  df-dom 8191  df-sdom 8192  df-fin 8193  df-fsupp 8512  df-oi 8651  df-cnf 8803
This theorem is referenced by:  cantnflem1d  8829  cantnflem1  8830  cantnflem3  8832
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