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Theorem cantnfp1 8940
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 6056 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
52, 3, 4syl2anc 576 . . . . . . . . . . . 12 (𝜑𝑋 ∈ On)
6 eloni 6041 . . . . . . . . . . . 12 (𝑋 ∈ On → Ord 𝑋)
7 ordirr 6049 . . . . . . . . . . . 12 (Ord 𝑋 → ¬ 𝑋𝑋)
85, 6, 73syl 18 . . . . . . . . . . 11 (𝜑 → ¬ 𝑋𝑋)
9 fvex 6514 . . . . . . . . . . . . . 14 (𝐺𝑋) ∈ V
10 dif1o 7929 . . . . . . . . . . . . . 14 ((𝐺𝑋) ∈ (V ∖ 1o) ↔ ((𝐺𝑋) ∈ V ∧ (𝐺𝑋) ≠ ∅))
119, 10mpbiran 696 . . . . . . . . . . . . 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 8925 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
1612, 15mpbid 224 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 487 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺:𝐵𝐴)
1817ffnd 6347 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 Fn 𝐵)
19 0ex 5069 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
2019a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ∅ ∈ V)
21 elsuppfn 7643 . . . . . . . . . . . . . . . . 17 ((𝐺 Fn 𝐵𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
2218, 2, 20, 21syl3anc 1351 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
2311bicomi 216 . . . . . . . . . . . . . . . . . 18 ((𝐺𝑋) ≠ ∅ ↔ (𝐺𝑋) ∈ (V ∖ 1o))
2423a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐺𝑋) ≠ ∅ ↔ (𝐺𝑋) ∈ (V ∖ 1o)))
2524anbi2d 619 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1o))))
2622, 25bitrd 271 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1o))))
27 cantnfp1.s . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
2827sseld 3859 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) → 𝑋𝑋))
2926, 28sylbird 252 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1o)) → 𝑋𝑋))
303, 29mpand 682 . . . . . . . . . . . . 13 (𝜑 → ((𝐺𝑋) ∈ (V ∖ 1o) → 𝑋𝑋))
3111, 30syl5bir 235 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑋) ≠ ∅ → 𝑋𝑋))
3231necon1bd 2985 . . . . . . . . . . 11 (𝜑 → (¬ 𝑋𝑋 → (𝐺𝑋) = ∅))
338, 32mpd 15 . . . . . . . . . 10 (𝜑 → (𝐺𝑋) = ∅)
3433ad3antrrr 717 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → (𝐺𝑋) = ∅)
35 simpr 477 . . . . . . . . . 10 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑡 = 𝑋)
3635fveq2d 6505 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → (𝐺𝑡) = (𝐺𝑋))
37 simpllr 763 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = ∅)
3834, 36, 373eqtr4rd 2825 . . . . . . . 8 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = (𝐺𝑡))
39 eqidd 2779 . . . . . . . 8 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ ¬ 𝑡 = 𝑋) → (𝐺𝑡) = (𝐺𝑡))
4038, 39ifeqda 4386 . . . . . . 7 (((𝜑𝑌 = ∅) ∧ 𝑡𝐵) → if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)) = (𝐺𝑡))
4140mpteq2dva 5023 . . . . . 6 ((𝜑𝑌 = ∅) → (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡))) = (𝑡𝐵 ↦ (𝐺𝑡)))
421, 41syl5eq 2826 . . . . 5 ((𝜑𝑌 = ∅) → 𝐹 = (𝑡𝐵 ↦ (𝐺𝑡)))
4317feqmptd 6564 . . . . . 6 (𝜑𝐺 = (𝑡𝐵 ↦ (𝐺𝑡)))
4443adantr 473 . . . . 5 ((𝜑𝑌 = ∅) → 𝐺 = (𝑡𝐵 ↦ (𝐺𝑡)))
4542, 44eqtr4d 2817 . . . 4 ((𝜑𝑌 = ∅) → 𝐹 = 𝐺)
4612adantr 473 . . . 4 ((𝜑𝑌 = ∅) → 𝐺𝑆)
4745, 46eqeltrd 2866 . . 3 ((𝜑𝑌 = ∅) → 𝐹𝑆)
48 oecl 7966 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
4914, 2, 48syl2anc 576 . . . . . . 7 (𝜑 → (𝐴o 𝐵) ∈ On)
5013, 14, 2cantnff 8933 . . . . . . . 8 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
5150, 12ffvelrnd 6679 . . . . . . 7 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴o 𝐵))
52 onelon 6056 . . . . . . 7 (((𝐴o 𝐵) ∈ On ∧ ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴o 𝐵)) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
5349, 51, 52syl2anc 576 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
5453adantr 473 . . . . 5 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
55 oa0r 7967 . . . . 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 6986 . . . . . 6 (𝑌 = ∅ → ((𝐴o 𝑋) ·o 𝑌) = ((𝐴o 𝑋) ·o ∅))
58 oecl 7966 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
5914, 5, 58syl2anc 576 . . . . . . 7 (𝜑 → (𝐴o 𝑋) ∈ On)
60 om0 7946 . . . . . . 7 ((𝐴o 𝑋) ∈ On → ((𝐴o 𝑋) ·o ∅) = ∅)
6159, 60syl 17 . . . . . 6 (𝜑 → ((𝐴o 𝑋) ·o ∅) = ∅)
6257, 61sylan9eqr 2836 . . . . 5 ((𝜑𝑌 = ∅) → ((𝐴o 𝑋) ·o 𝑌) = ∅)
6362oveq1d 6993 . . . 4 ((𝜑𝑌 = ∅) → (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)) = (∅ +o ((𝐴 CNF 𝐵)‘𝐺)))
6445fveq2d 6505 . . . 4 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = ((𝐴 CNF 𝐵)‘𝐺))
6556, 63, 643eqtr4rd 2825 . . 3 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
6647, 65jca 504 . 2 ((𝜑𝑌 = ∅) → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
6714adantr 473 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐴 ∈ On)
682adantr 473 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐵 ∈ On)
6912adantr 473 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐺𝑆)
703adantr 473 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝑋𝐵)
71 cantnfp1.y . . . . 5 (𝜑𝑌𝐴)
7271adantr 473 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝑌𝐴)
7327adantr 473 . . . 4 ((𝜑𝑌 ≠ ∅) → (𝐺 supp ∅) ⊆ 𝑋)
7413, 67, 68, 69, 70, 72, 73, 1cantnfp1lem1 8937 . . 3 ((𝜑𝑌 ≠ ∅) → 𝐹𝑆)
75 onelon 6056 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
7614, 71, 75syl2anc 576 . . . . . 6 (𝜑𝑌 ∈ On)
77 on0eln0 6086 . . . . . 6 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
7876, 77syl 17 . . . . 5 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
7978biimpar 470 . . . 4 ((𝜑𝑌 ≠ ∅) → ∅ ∈ 𝑌)
80 eqid 2778 . . . 4 OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅))
81 eqid 2778 . . . 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 2778 . . . 4 OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
83 eqid 2778 . . . 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 8939 . . 3 ((𝜑𝑌 ≠ ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
8574, 84jca 504 . 2 ((𝜑𝑌 ≠ ∅) → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
8666, 85pm2.61dane 3055 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 198  wa 387   = wceq 1507  wcel 2050  wne 2967  Vcvv 3415  cdif 3828  wss 3831  c0 4180  ifcif 4351   class class class wbr 4930  cmpt 5009   E cep 5317  dom cdm 5408  Ord word 6030  Oncon0 6031   Fn wfn 6185  wf 6186  cfv 6190  (class class class)co 6978  cmpo 6980   supp csupp 7635  seq𝜔cseqom 7888  1oc1o 7900   +o coa 7904   ·o comu 7905  o coe 7906   finSupp cfsupp 8630  OrdIsocoi 8770   CNF ccnf 8920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-se 5368  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-isom 6199  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-1st 7503  df-2nd 7504  df-supp 7636  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-seqom 7889  df-1o 7907  df-2o 7908  df-oadd 7911  df-omul 7912  df-oexp 7913  df-er 8091  df-map 8210  df-en 8309  df-dom 8310  df-sdom 8311  df-fin 8312  df-fsupp 8631  df-oi 8771  df-cnf 8921
This theorem is referenced by:  cantnflem1d  8947  cantnflem1  8948  cantnflem3  8950
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