| Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > 1t1e1ALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of 1t1e1 12428 using a different set of axioms (add ax-mulrcl 11218, ax-i2m1 11223, ax-1ne0 11224, ax-rrecex 11227 and remove ax-resscn 11212, ax-mulcom 11219, ax-mulass 11221, ax-distr 11222). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| 1t1e1ALT | ⊢ (1 · 1) = 1 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1re 11261 | . 2 ⊢ 1 ∈ ℝ | |
| 2 | ax-1rid 11225 | . 2 ⊢ (1 ∈ ℝ → (1 · 1) = 1) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (1 · 1) = 1 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℝcr 11154 1c1 11156 · cmul 11160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-mulrcl 11218 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rrecex 11227 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: nnmul1com 42306 remulinvcom 42462 sn-0tie0 42469 |
| Copyright terms: Public domain | W3C validator |