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Theorem omndmul2 31920
Description: In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
Hypotheses
Ref Expression
omndmul.0 𝐵 = (Base‘𝑀)
omndmul.1 = (le‘𝑀)
omndmul2.2 · = (.g𝑀)
omndmul2.3 0 = (0g𝑀)
Assertion
Ref Expression
omndmul2 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))

Proof of Theorem omndmul2
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-3an 1089 . . 3 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)) ∧ 0 𝑋))
2 anass 469 . . . 4 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ↔ (𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)))
32anbi1i 624 . . 3 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)) ∧ 0 𝑋))
41, 3bitr4i 277 . 2 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) ↔ (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋))
5 simplr 767 . . 3 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 𝑁 ∈ ℕ0)
6 oveq1 7364 . . . . 5 (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
76breq2d 5117 . . . 4 (𝑚 = 0 → ( 0 (𝑚 · 𝑋) ↔ 0 (0 · 𝑋)))
8 oveq1 7364 . . . . 5 (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
98breq2d 5117 . . . 4 (𝑚 = 𝑛 → ( 0 (𝑚 · 𝑋) ↔ 0 (𝑛 · 𝑋)))
10 oveq1 7364 . . . . 5 (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
1110breq2d 5117 . . . 4 (𝑚 = (𝑛 + 1) → ( 0 (𝑚 · 𝑋) ↔ 0 ((𝑛 + 1) · 𝑋)))
12 oveq1 7364 . . . . 5 (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
1312breq2d 5117 . . . 4 (𝑚 = 𝑁 → ( 0 (𝑚 · 𝑋) ↔ 0 (𝑁 · 𝑋)))
14 omndtos 31913 . . . . . . . 8 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
15 tospos 18309 . . . . . . . 8 (𝑀 ∈ Toset → 𝑀 ∈ Poset)
1614, 15syl 17 . . . . . . 7 (𝑀 ∈ oMnd → 𝑀 ∈ Poset)
17 omndmnd 31912 . . . . . . . 8 (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
18 omndmul.0 . . . . . . . . 9 𝐵 = (Base‘𝑀)
19 omndmul2.3 . . . . . . . . 9 0 = (0g𝑀)
2018, 19mndidcl 18571 . . . . . . . 8 (𝑀 ∈ Mnd → 0𝐵)
2117, 20syl 17 . . . . . . 7 (𝑀 ∈ oMnd → 0𝐵)
22 omndmul.1 . . . . . . . 8 = (le‘𝑀)
2318, 22posref 18207 . . . . . . 7 ((𝑀 ∈ Poset ∧ 0𝐵) → 0 0 )
2416, 21, 23syl2anc 584 . . . . . 6 (𝑀 ∈ oMnd → 0 0 )
2524ad3antrrr 728 . . . . 5 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 0 )
26 omndmul2.2 . . . . . . 7 · = (.g𝑀)
2718, 19, 26mulg0 18879 . . . . . 6 (𝑋𝐵 → (0 · 𝑋) = 0 )
2827ad3antlr 729 . . . . 5 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → (0 · 𝑋) = 0 )
2925, 28breqtrrd 5133 . . . 4 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (0 · 𝑋))
3016ad5antr 732 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑀 ∈ Poset)
3117ad5antr 732 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑀 ∈ Mnd)
3231, 20syl 17 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0𝐵)
33 simplr 767 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑛 ∈ ℕ0)
34 simp-5r 784 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑋𝐵)
3518, 26, 31, 33, 34mulgnn0cld 18897 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 · 𝑋) ∈ 𝐵)
36 simpr32 1264 . . . . . . . . . 10 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → 𝑛 ∈ ℕ0)
37 1nn0 12429 . . . . . . . . . . 11 1 ∈ ℕ0
3837a1i 11 . . . . . . . . . 10 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → 1 ∈ ℕ0)
3936, 38nn0addcld 12477 . . . . . . . . 9 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → (𝑛 + 1) ∈ ℕ0)
40393anassrs 1360 . . . . . . . 8 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋))) → (𝑛 + 1) ∈ ℕ0)
41403anassrs 1360 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 + 1) ∈ ℕ0)
4218, 26, 31, 41, 34mulgnn0cld 18897 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
4332, 35, 423jca 1128 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → ( 0𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵))
44 simpr 485 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0 (𝑛 · 𝑋))
45 simp-4l 781 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ oMnd)
4617ad4antr 730 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ Mnd)
4746, 20syl 17 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 0𝐵)
48 simp-4r 782 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋𝐵)
49 simpr 485 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
5018, 26, 46, 49, 48mulgnn0cld 18897 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ∈ 𝐵)
51 simplr 767 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 0 𝑋)
52 eqid 2736 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
5318, 22, 52omndadd 31914 . . . . . . . 8 ((𝑀 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝑀)(𝑛 · 𝑋)) (𝑋(+g𝑀)(𝑛 · 𝑋)))
5445, 47, 48, 50, 51, 53syl131anc 1383 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g𝑀)(𝑛 · 𝑋)) (𝑋(+g𝑀)(𝑛 · 𝑋)))
5518, 52, 19mndlid 18576 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑛 · 𝑋) ∈ 𝐵) → ( 0 (+g𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
5646, 50, 55syl2anc 584 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
5737a1i 11 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 1 ∈ ℕ0)
5818, 26, 52mulgnn0dir 18906 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (1 ∈ ℕ0𝑛 ∈ ℕ0𝑋𝐵)) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)))
5946, 57, 49, 48, 58syl13anc 1372 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)))
60 1cnd 11150 . . . . . . . . . . 11 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 1 ∈ ℂ)
61 simpr3 1196 . . . . . . . . . . . 12 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 𝑛 ∈ ℕ0)
6261nn0cnd 12475 . . . . . . . . . . 11 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 𝑛 ∈ ℂ)
6360, 62addcomd 11357 . . . . . . . . . 10 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → (1 + 𝑛) = (𝑛 + 1))
64633anassrs 1360 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 + 𝑛) = (𝑛 + 1))
6564oveq1d 7372 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((𝑛 + 1) · 𝑋))
6618, 26mulg1 18883 . . . . . . . . . 10 (𝑋𝐵 → (1 · 𝑋) = 𝑋)
6748, 66syl 17 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 · 𝑋) = 𝑋)
6867oveq1d 7372 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)) = (𝑋(+g𝑀)(𝑛 · 𝑋)))
6959, 65, 683eqtr3rd 2785 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑋(+g𝑀)(𝑛 · 𝑋)) = ((𝑛 + 1) · 𝑋))
7054, 56, 693brtr3d 5136 . . . . . 6 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋))
7170adantr 481 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋))
7218, 22postr 18209 . . . . . 6 ((𝑀 ∈ Poset ∧ ( 0𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵)) → (( 0 (𝑛 · 𝑋) ∧ (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋)) → 0 ((𝑛 + 1) · 𝑋)))
7372imp 407 . . . . 5 (((𝑀 ∈ Poset ∧ ( 0𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵)) ∧ ( 0 (𝑛 · 𝑋) ∧ (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋))) → 0 ((𝑛 + 1) · 𝑋))
7430, 43, 44, 71, 73syl22anc 837 . . . 4 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0 ((𝑛 + 1) · 𝑋))
757, 9, 11, 13, 29, 74nn0indd 12600 . . 3 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑁 ∈ ℕ0) → 0 (𝑁 · 𝑋))
765, 75mpdan 685 . 2 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))
774, 76sylbi 216 1 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5105  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054  0cn0 12413  Basecbs 17083  +gcplusg 17133  lecple 17140  0gc0g 17321  Posetcpo 18196  Tosetctos 18305  Mndcmnd 18556  .gcmg 18872  oMndcomnd 31905
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907  df-0g 17323  df-proset 18184  df-poset 18202  df-toset 18306  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mulg 18873  df-omnd 31907
This theorem is referenced by:  omndmul3  31921
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