Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cantnfp1 Structured version   Visualization version   GIF version

Theorem cantnfp1 9128
 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 6184 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑋𝐵) → 𝑋 ∈ On)
52, 3, 4syl2anc 587 . . . . . . . . . . . 12 (𝜑𝑋 ∈ On)
6 eloni 6169 . . . . . . . . . . . 12 (𝑋 ∈ On → Ord 𝑋)
7 ordirr 6177 . . . . . . . . . . . 12 (Ord 𝑋 → ¬ 𝑋𝑋)
85, 6, 73syl 18 . . . . . . . . . . 11 (𝜑 → ¬ 𝑋𝑋)
9 fvex 6658 . . . . . . . . . . . . . 14 (𝐺𝑋) ∈ V
10 dif1o 8108 . . . . . . . . . . . . . 14 ((𝐺𝑋) ∈ (V ∖ 1o) ↔ ((𝐺𝑋) ∈ V ∧ (𝐺𝑋) ≠ ∅))
119, 10mpbiran 708 . . . . . . . . . . . . 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 9113 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐺𝑆 ↔ (𝐺:𝐵𝐴𝐺 finSupp ∅)))
1612, 15mpbid 235 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐺:𝐵𝐴𝐺 finSupp ∅))
1716simpld 498 . . . . . . . . . . . . . . . . . 18 (𝜑𝐺:𝐵𝐴)
1817ffnd 6488 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 Fn 𝐵)
19 0ex 5175 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
2019a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ∅ ∈ V)
21 elsuppfn 7821 . . . . . . . . . . . . . . . . 17 ((𝐺 Fn 𝐵𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
2218, 2, 20, 21syl3anc 1368 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅)))
2311bicomi 227 . . . . . . . . . . . . . . . . . 18 ((𝐺𝑋) ≠ ∅ ↔ (𝐺𝑋) ∈ (V ∖ 1o))
2423a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐺𝑋) ≠ ∅ ↔ (𝐺𝑋) ∈ (V ∖ 1o)))
2524anbi2d 631 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑋𝐵 ∧ (𝐺𝑋) ≠ ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1o))))
2622, 25bitrd 282 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1o))))
27 cantnfp1.s . . . . . . . . . . . . . . . 16 (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
2827sseld 3914 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 ∈ (𝐺 supp ∅) → 𝑋𝑋))
2926, 28sylbird 263 . . . . . . . . . . . . . 14 (𝜑 → ((𝑋𝐵 ∧ (𝐺𝑋) ∈ (V ∖ 1o)) → 𝑋𝑋))
303, 29mpand 694 . . . . . . . . . . . . 13 (𝜑 → ((𝐺𝑋) ∈ (V ∖ 1o) → 𝑋𝑋))
3111, 30syl5bir 246 . . . . . . . . . . . 12 (𝜑 → ((𝐺𝑋) ≠ ∅ → 𝑋𝑋))
3231necon1bd 3005 . . . . . . . . . . 11 (𝜑 → (¬ 𝑋𝑋 → (𝐺𝑋) = ∅))
338, 32mpd 15 . . . . . . . . . 10 (𝜑 → (𝐺𝑋) = ∅)
3433ad3antrrr 729 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → (𝐺𝑋) = ∅)
35 simpr 488 . . . . . . . . . 10 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑡 = 𝑋)
3635fveq2d 6649 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → (𝐺𝑡) = (𝐺𝑋))
37 simpllr 775 . . . . . . . . 9 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = ∅)
3834, 36, 373eqtr4rd 2844 . . . . . . . 8 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = (𝐺𝑡))
39 eqidd 2799 . . . . . . . 8 ((((𝜑𝑌 = ∅) ∧ 𝑡𝐵) ∧ ¬ 𝑡 = 𝑋) → (𝐺𝑡) = (𝐺𝑡))
4038, 39ifeqda 4460 . . . . . . 7 (((𝜑𝑌 = ∅) ∧ 𝑡𝐵) → if(𝑡 = 𝑋, 𝑌, (𝐺𝑡)) = (𝐺𝑡))
4140mpteq2dva 5125 . . . . . 6 ((𝜑𝑌 = ∅) → (𝑡𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺𝑡))) = (𝑡𝐵 ↦ (𝐺𝑡)))
421, 41syl5eq 2845 . . . . 5 ((𝜑𝑌 = ∅) → 𝐹 = (𝑡𝐵 ↦ (𝐺𝑡)))
4317feqmptd 6708 . . . . . 6 (𝜑𝐺 = (𝑡𝐵 ↦ (𝐺𝑡)))
4443adantr 484 . . . . 5 ((𝜑𝑌 = ∅) → 𝐺 = (𝑡𝐵 ↦ (𝐺𝑡)))
4542, 44eqtr4d 2836 . . . 4 ((𝜑𝑌 = ∅) → 𝐹 = 𝐺)
4612adantr 484 . . . 4 ((𝜑𝑌 = ∅) → 𝐺𝑆)
4745, 46eqeltrd 2890 . . 3 ((𝜑𝑌 = ∅) → 𝐹𝑆)
48 oecl 8145 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
4914, 2, 48syl2anc 587 . . . . . . 7 (𝜑 → (𝐴o 𝐵) ∈ On)
5013, 14, 2cantnff 9121 . . . . . . . 8 (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴o 𝐵))
5150, 12ffvelrnd 6829 . . . . . . 7 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴o 𝐵))
52 onelon 6184 . . . . . . 7 (((𝐴o 𝐵) ∈ On ∧ ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴o 𝐵)) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
5349, 51, 52syl2anc 587 . . . . . 6 (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
5453adantr 484 . . . . 5 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
55 oa0r 8146 . . . . 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 7143 . . . . . 6 (𝑌 = ∅ → ((𝐴o 𝑋) ·o 𝑌) = ((𝐴o 𝑋) ·o ∅))
58 oecl 8145 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴o 𝑋) ∈ On)
5914, 5, 58syl2anc 587 . . . . . . 7 (𝜑 → (𝐴o 𝑋) ∈ On)
60 om0 8125 . . . . . . 7 ((𝐴o 𝑋) ∈ On → ((𝐴o 𝑋) ·o ∅) = ∅)
6159, 60syl 17 . . . . . 6 (𝜑 → ((𝐴o 𝑋) ·o ∅) = ∅)
6257, 61sylan9eqr 2855 . . . . 5 ((𝜑𝑌 = ∅) → ((𝐴o 𝑋) ·o 𝑌) = ∅)
6362oveq1d 7150 . . . 4 ((𝜑𝑌 = ∅) → (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)) = (∅ +o ((𝐴 CNF 𝐵)‘𝐺)))
6445fveq2d 6649 . . . 4 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = ((𝐴 CNF 𝐵)‘𝐺))
6556, 63, 643eqtr4rd 2844 . . 3 ((𝜑𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
6647, 65jca 515 . 2 ((𝜑𝑌 = ∅) → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
6714adantr 484 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐴 ∈ On)
682adantr 484 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐵 ∈ On)
6912adantr 484 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝐺𝑆)
703adantr 484 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝑋𝐵)
71 cantnfp1.y . . . . 5 (𝜑𝑌𝐴)
7271adantr 484 . . . 4 ((𝜑𝑌 ≠ ∅) → 𝑌𝐴)
7327adantr 484 . . . 4 ((𝜑𝑌 ≠ ∅) → (𝐺 supp ∅) ⊆ 𝑋)
7413, 67, 68, 69, 70, 72, 73, 1cantnfp1lem1 9125 . . 3 ((𝜑𝑌 ≠ ∅) → 𝐹𝑆)
75 onelon 6184 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑌𝐴) → 𝑌 ∈ On)
7614, 71, 75syl2anc 587 . . . . . 6 (𝜑𝑌 ∈ On)
77 on0eln0 6214 . . . . . 6 (𝑌 ∈ On → (∅ ∈ 𝑌𝑌 ≠ ∅))
7876, 77syl 17 . . . . 5 (𝜑 → (∅ ∈ 𝑌𝑌 ≠ ∅))
7978biimpar 481 . . . 4 ((𝜑𝑌 ≠ ∅) → ∅ ∈ 𝑌)
80 eqid 2798 . . . 4 OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅))
81 eqid 2798 . . . 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 2798 . . . 4 OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
83 eqid 2798 . . . 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 9127 . . 3 ((𝜑𝑌 ≠ ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
8574, 84jca 515 . 2 ((𝜑𝑌 ≠ ∅) → (𝐹𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
8666, 85pm2.61dane 3074 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  ifcif 4425   class class class wbr 5030   ↦ cmpt 5110   E cep 5429  dom cdm 5519  Ord word 6158  Oncon0 6159   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   supp csupp 7813  seqωcseqom 8066  1oc1o 8078   +o coa 8082   ·o comu 8083   ↑o coe 8084   finSupp cfsupp 8817  OrdIsocoi 8957   CNF ccnf 9108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-seqom 8067  df-1o 8085  df-2o 8086  df-oadd 8089  df-omul 8090  df-oexp 8091  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-oi 8958  df-cnf 9109 This theorem is referenced by:  cantnflem1d  9135  cantnflem1  9136  cantnflem3  9138
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