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Theorem naddcom 8629
Description: Natural addition commutes. (Contributed by Scott Fenton, 26-Aug-2024.)
Assertion
Ref Expression
naddcom ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))

Proof of Theorem naddcom
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7365 . . 3 (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
2 oveq2 7366 . . 3 (𝑎 = 𝑐 → (𝑏 +no 𝑎) = (𝑏 +no 𝑐))
31, 2eqeq12d 2753 . 2 (𝑎 = 𝑐 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝑐 +no 𝑏) = (𝑏 +no 𝑐)))
4 oveq2 7366 . . 3 (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
5 oveq1 7365 . . 3 (𝑏 = 𝑑 → (𝑏 +no 𝑐) = (𝑑 +no 𝑐))
64, 5eqeq12d 2753 . 2 (𝑏 = 𝑑 → ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
7 oveq1 7365 . . 3 (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
8 oveq2 7366 . . 3 (𝑎 = 𝑐 → (𝑑 +no 𝑎) = (𝑑 +no 𝑐))
97, 8eqeq12d 2753 . 2 (𝑎 = 𝑐 → ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) ↔ (𝑐 +no 𝑑) = (𝑑 +no 𝑐)))
10 oveq1 7365 . . 3 (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
11 oveq2 7366 . . 3 (𝑎 = 𝐴 → (𝑏 +no 𝑎) = (𝑏 +no 𝐴))
1210, 11eqeq12d 2753 . 2 (𝑎 = 𝐴 → ((𝑎 +no 𝑏) = (𝑏 +no 𝑎) ↔ (𝐴 +no 𝑏) = (𝑏 +no 𝐴)))
13 oveq2 7366 . . 3 (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
14 oveq1 7365 . . 3 (𝑏 = 𝐵 → (𝑏 +no 𝐴) = (𝐵 +no 𝐴))
1513, 14eqeq12d 2753 . 2 (𝑏 = 𝐵 → ((𝐴 +no 𝑏) = (𝑏 +no 𝐴) ↔ (𝐴 +no 𝐵) = (𝐵 +no 𝐴)))
16 eleq1 2826 . . . . . . . . . . . 12 ((𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
1716ralimi 3087 . . . . . . . . . . 11 (∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → ∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥))
18 ralbi 3107 . . . . . . . . . . 11 (∀𝑑𝑏 ((𝑎 +no 𝑑) ∈ 𝑥 ↔ (𝑑 +no 𝑎) ∈ 𝑥) → (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
1917, 18syl 17 . . . . . . . . . 10 (∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎) → (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
20193ad2ant3 1136 . . . . . . . . 9 ((∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
2120adantl 483 . . . . . . . 8 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ↔ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥))
22 eleq1 2826 . . . . . . . . . . . 12 ((𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
2322ralimi 3087 . . . . . . . . . . 11 (∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → ∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥))
24 ralbi 3107 . . . . . . . . . . 11 (∀𝑐𝑎 ((𝑐 +no 𝑏) ∈ 𝑥 ↔ (𝑏 +no 𝑐) ∈ 𝑥) → (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
2523, 24syl 17 . . . . . . . . . 10 (∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) → (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
26253ad2ant2 1135 . . . . . . . . 9 ((∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
2726adantl 483 . . . . . . . 8 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥 ↔ ∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥))
2821, 27anbi12d 632 . . . . . . 7 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → ((∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥) ↔ (∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥)))
2928biancomd 465 . . . . . 6 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → ((∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥) ↔ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)))
3029rabbidv 3416 . . . . 5 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → {𝑥 ∈ On ∣ (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥)} = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
3130inteqd 4913 . . . 4 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → {𝑥 ∈ On ∣ (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥)} = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
32 naddov2 8626 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥)})
3332adantr 482 . . . 4 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑎 +no 𝑏) = {𝑥 ∈ On ∣ (∀𝑑𝑏 (𝑎 +no 𝑑) ∈ 𝑥 ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) ∈ 𝑥)})
34 naddov2 8626 . . . . . 6 ((𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑏 +no 𝑎) = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
3534ancoms 460 . . . . 5 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 +no 𝑎) = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
3635adantr 482 . . . 4 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑏 +no 𝑎) = {𝑥 ∈ On ∣ (∀𝑐𝑎 (𝑏 +no 𝑐) ∈ 𝑥 ∧ ∀𝑑𝑏 (𝑑 +no 𝑎) ∈ 𝑥)})
3731, 33, 363eqtr4d 2787 . . 3 (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ (∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎))) → (𝑎 +no 𝑏) = (𝑏 +no 𝑎))
3837ex 414 . 2 ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐𝑎𝑑𝑏 (𝑐 +no 𝑑) = (𝑑 +no 𝑐) ∧ ∀𝑐𝑎 (𝑐 +no 𝑏) = (𝑏 +no 𝑐) ∧ ∀𝑑𝑏 (𝑎 +no 𝑑) = (𝑑 +no 𝑎)) → (𝑎 +no 𝑏) = (𝑏 +no 𝑎)))
393, 6, 9, 12, 15, 38on2ind 8616 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065  {crab 3408   cint 4908  Oncon0 6318  (class class class)co 7358   +no cnadd 8612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-frecs 8213  df-nadd 8613
This theorem is referenced by:  naddel2  8634  naddss2  8636  nadd32  8642  nadd42  8644  addsproplem2  27285
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